{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "1581ad7f-f155-4170-89ab-67dae15a895c",
   "metadata": {},
   "source": [
    "# Práctica 7"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79fc85d5-9ee1-42ee-921e-0d236fd46e2a",
   "metadata": {},
   "source": [
    "## Ejercicio 1\n",
    "\n",
    "Se analizaron 12 piezas de pan blanco de cierta marca elegidas al azar. Se determinó el porcentaje de carbohidratos contenido en cada una de las piezas, obteniéndose los siguientes valores:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d63f957c-8ef1-4f13-8aee-814dd4fff011",
   "metadata": {},
   "outputs": [],
   "source": [
    "muestra = c(76.93, 76.88, 77.07, 76.68, 76.39, 75.09, 77.67, 76.88, 78.15, 76.50, 77.16, 76.42)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0dff5d1-6ced-4ad4-bf35-c4174e4c4185",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Estimar la esperanza del porcentaje de carbohidratos contenido en una pieza de pan de esta marca."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3272a824-82b8-4712-9b05-40d778900c3c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "76.8183333333333"
      ],
      "text/latex": [
       "76.8183333333333"
      ],
      "text/markdown": [
       "76.8183333333333"
      ],
      "text/plain": [
       "[1] 76.81833"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mean(muestra)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4bfe3577-afc1-4c79-af21-6272efe7c343",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Estimar la mediana del porcentaje de carbohidratos contenido en una pieza de pan de esta marca."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3c22a2e9-b60c-4eb7-8bc8-d074cdfe2b3a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "76.88"
      ],
      "text/latex": [
       "76.88"
      ],
      "text/markdown": [
       "76.88"
      ],
      "text/plain": [
       "[1] 76.88"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "median(muestra)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8575668b-4038-40f7-976d-323668107095",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Estimar la probabilidad de que el porcentaje de carbohidratos de una pieza de pan de esta marca no exceda el 76.5%."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f5fcb56b-17de-45a2-aad8-a662886d4fc5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.333333333333333"
      ],
      "text/latex": [
       "0.333333333333333"
      ],
      "text/markdown": [
       "0.333333333333333"
      ],
      "text/plain": [
       "[1] 0.3333333"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum(muestra <= 76.5) / length(muestra)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82d51d77-4413-409d-865a-77f129ff586d",
   "metadata": {},
   "source": [
    "## Ejercicio 2\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria de una distribución $\\mathcal{N}(\\mu, \\sigma^2)$. Obtener los estimadores de máxima verosimilitud (MV) de:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a826e56a-55bf-4a49-afae-be683d2b092e",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "El par $(\\mu, \\sigma^2)$ simultáneamente.\n",
    "\n",
    "Podemos directamente plantear las fórmulas correspondientes para los parámetros $\\mu$ y $\\sigma$ de una distribución normal. Sino, usamos la definición de los EMV, planteamos un sistema de 2 ecuaciones (1 para cada parámetro) y luego despejamos. Eso sería la demostración de cómo se llegan a estas fórmulas.\n",
    "\n",
    "$\\hat{\\mu} = \\bar{X}$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\\hat{\\sigma}^2 = \\frac{\\sum_{i=1}^{n} (X_i - \\bar{X})^2}{n}\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "*Nota: En el apunte de Bianco y Martínez (2004), página 166, está el desarrollo completo de cómo llegar a estas fórmulas.*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "562259e2-5f1a-4958-a482-c8b6cc83a87d",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "$\\mu$, siendo $\\sigma^2 = \\sigma^2_0$ conocida.\n",
    "\n",
    "Como $\\sigma$ es conocida, solo estimamos $\\mu$.\n",
    "\n",
    "$\\hat{\\mu} = \\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06a4ea36-0d70-457d-bd6f-4aa0efa141b6",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "$\\sigma^2$, siendo $\\mu = \\mu_0$ conocida.\n",
    "\n",
    "De forma análoga, ahora solo estimamos $\\sigma$. Observar que el estimador para $\\mu$ no depende de $\\sigma$, mientras que en este caso, el estimador de $\\sigma$ sí depende de $\\mu$. Por lo tanto, no hay que usar tal cual las fórmulas genéricas de los estimadores, sino retroceder algunos pasos antes de reemplazar $\\mu$ por su estimador y en cambio reemplazarlo por $\\mu_0$.\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\\hat{\\sigma}^2 = \\frac{\\sum_{i=1}^{n} (X_i - \\mu_0)^2}{n}\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34e3d504-9593-49b3-b422-171b44b789fe",
   "metadata": {},
   "source": [
    "## Ejercicio 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd5084e4-c031-4aaf-bf5a-e715f8ff0901",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Una máquina envasa caramelos, siendo el peso neto (en gramos) de cada bolsa una v.a. con distribución normal. Los siguientes datos corresponden al peso de 15 bolsas elegidas al azar:\n",
    "\n",
    "```\n",
    "210 197 187 217 194 208 220 199 193 203 181 212 188 196 185\n",
    "```\n",
    "\n",
    "Usando el método de MV, indique el valor estimado de la media y la varianza del peso neto de cada bolsa de caramelos envasada por esta máquina."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e5dff432-246f-43bb-8c5b-e0ebb1e840f0",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = c(210, 197, 187, 217, 194, 208, 220, 199, 193, 203, 181, 212, 188, 196, 185)\n",
    "n = length(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "4459f45c-e1cb-42e3-81b9-bbcd63c58db6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "199.333333333333"
      ],
      "text/latex": [
       "199.333333333333"
      ],
      "text/markdown": [
       "199.333333333333"
      ],
      "text/plain": [
       "[1] 199.3333"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mu = mean(X)\n",
    "mu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "613bdac8-2552-41f5-8b1f-783b6555a557",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "133.955555555556"
      ],
      "text/latex": [
       "133.955555555556"
      ],
      "text/markdown": [
       "133.955555555556"
      ],
      "text/plain": [
       "[1] 133.9556"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sigma = sqrt(sum((X - mean(X))^2) / n)\n",
    "sigma^2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7a4e4c2-874e-483f-bf57-65938c87db2d",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Con cierto instrumento se realizan 20 mediciones de una magnitud física $\\mu$. Cada observación es de la forma $X = \\mu + \\epsilon$, donde $\\epsilon$ es el error de medición (aleatorio). Se obtuvieron los siguientes valores:\n",
    "\n",
    "```\n",
    "25.11 25.02 25.16 24.98 24.83 25.05 24.94 25.04 24.99 24.96 25.03 24.97 24.93 25.12 25.01 25.12 24.90 24.98 25.10 24.96\n",
    "```\n",
    "\n",
    "Suponiendo que los errores de medición tienen distribución normal con media cero y varianza 0.01, estimar $\\mu$. ¿Cuál es la varianza del estimador de $\\mu$?\n",
    "\n",
    "$\\epsilon \\sim \\mathcal{N}(0, 0.01)$\n",
    "\n",
    "$X = \\mu + \\epsilon \\implies X \\sim \\mathcal{N}(\\mu, 0.01)$\n",
    "\n",
    "No se exactamente cuál es la justificación 100% formal de esto último, pero como en este contexto $\\mu$ representa una magnitud física, en esencia es una constante que se suma al error de medición $\\epsilon$. En efecto, esto \"desfaza\" de forma constante la distribución normal que trae $\\epsilon$ generando así la v.a. $X$ que consecuentemente también tiene distribución normal, solo que desfazada respecto a la de $\\epsilon$. A su vez, como la esperanza de $\\epsilon$ era cero, la esperanza de $X$ termina siendo la constante $\\mu$. Mientras que la varianza se mentiene igual a la de $\\epsilon$ ya que la suma de constantes no afecta a la varianza.\n",
    "\n",
    "Dicho todo esto, ahora podemos estimar $\\mu$ usando la fórmula correspondiente para una distribución normal.\n",
    "\n",
    "$\\hat{\\mu} = \\bar{X}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "d953abda-fdcd-4f64-8b17-ea1067e6e5be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "25.01"
      ],
      "text/latex": [
       "25.01"
      ],
      "text/markdown": [
       "25.01"
      ],
      "text/plain": [
       "[1] 25.01"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = c(25.11, 25.02, 25.16, 24.98, 24.83, 25.05, 24.94, 25.04, 24.99, 24.96, 25.03, 24.97, 24.93, 25.12, 25.01, 25.12, 24.90, 24.98, 25.10, 24.96)\n",
    "mu = mean(X)\n",
    "mu"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03a49cc9-8d17-4c62-8a11-aff0fb700e14",
   "metadata": {},
   "source": [
    "Para calcular la varianza del estimador de $\\mu$ podemos hacer lo siguiente.\n",
    "\n",
    "Recordar que: $V(\\mu + \\epsilon_i) = V(\\epsilon_i) = 0.01$ (pues $\\mu$ es una constante y todas las $\\epsilon$ son i.i.d.)\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "V(\\hat{\\mu})\n",
    "= V(\\bar{X})\n",
    "= V\\Big(\\frac{\\sum_{i=1}^{20} X_i}{20}\\Big)\n",
    "= V\\Big(\\frac{\\sum_{i=1}^{20} \\mu + \\epsilon_i}{20}\\Big)\n",
    "= \\frac{1}{20^2} \\sum_{i=1}^{20} V(\\mu + \\epsilon_i)\n",
    "= \\frac{20 * 0.01}{20^2}\n",
    "= \\frac{0.01}{20}\n",
    "= 0.0005\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "996cdb70-3ef8-4d22-9dae-34a2fb245524",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Para conocer la precisión de un sistema de medición se mide 24 veces una magnitud conocida $\\mu_0 = 12$, obteniéndose los siguientes valores:\n",
    "\n",
    "```\n",
    "12.51 11.66 11.91 12.25 11.54 11.36 12.40 12.19 12.88 12.16 12.69 12.91 12.12 11.02 12.53 11.77 12.72 10.56 11.52 11.66 12.25 12.09 11.48 12.36\n",
    "```\n",
    "\n",
    "Estimar la precisión (es decir, la varianza del error de medición), suponiendo que los errores están normalmente distribuídos con media cero."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f76c0fb3-61b9-4d06-a9c4-82a8153cbbc0",
   "metadata": {},
   "source": [
    "Sea $\\epsilon \\sim \\mathcal{N}(0, \\sigma^2)$ el error de una medición.\n",
    "\n",
    "Sea $X = \\mu_0 + \\epsilon$ el valor de una medición.\n",
    "\n",
    "$X \\sim \\mathcal{N}(\\mu_0, \\sigma^2)$ (por la misma justificación que la pregunta anterior)\n",
    "\n",
    "Ahora usamos la fórmula para estimar $\\sigma$ de una distribución normal.\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\\hat{\\sigma}^2 = \\frac{\\sum_{i=1}^{24} (X_i - \\mu_0)^2}{24}\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "d1b96784-d26a-42e7-b150-43bd93152f84",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.336875"
      ],
      "text/latex": [
       "0.336875"
      ],
      "text/markdown": [
       "0.336875"
      ],
      "text/plain": [
       "[1] 0.336875"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = c(12.51, 11.66, 11.91, 12.25, 11.54, 11.36, 12.40, 12.19, 12.88, 12.16, 12.69, 12.91, 12.12, 11.02, 12.53, 11.77, 12.72, 10.56, 11.52, 11.66, 12.25, 12.09, 11.48, 12.36)\n",
    "n = length(X)\n",
    "mu0 = 12\n",
    "var = sum((X - mu0)^2) / n\n",
    "var"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96250cc1-5b71-48c4-bd81-42d892125534",
   "metadata": {},
   "source": [
    "## Ejercicio 4\n",
    "\n",
    "Se sabe que el tiempo de duración de una clase de lámparas tiene distribución $\\mathcal{E}(\\theta)$. Consideremos el siguiente experimento. Se seleccionan $n$ lámparas al azar y se registran sus tiempos de duración: $X_1, \\dots, X_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b2132d7-0a44-4f67-b287-ec9a8587a9c7",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Calcular el estimador de momentos y de MV de $\\theta$, y también del percentil 90 (o cuantil 0.9) del tiempo de duración de una lámpara bajo el modelo propuesto.\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\bar{X} = E(X) \\iff \\bar{X} = \\tfrac{1}{\\theta} \\implies \\hat{\\theta} = \\tfrac{1}{\\bar{X}}$\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\\bar{X} = E(X) \\iff \\bar{X} = \\tfrac{1}{\\theta} \\implies \\hat{\\theta} = \\tfrac{1}{\\bar{X}}$\n",
    "\n",
    "*Nota: En la distribución exponencial coinciden ambos estimadores.*\n",
    "\n",
    "**Percentil 90**\n",
    "\n",
    "$1 - e^{-\\theta x} = 0.9 \\iff x = \\tfrac{-ln(0.1)}{\\theta} = \\tfrac{-ln(0.1)}{\\tfrac{1}{\\bar{X}}} = -ln(0.1) \\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "838fc2fd-00c9-435e-9c9b-cafeb6f0c7a0",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Decidir si estos estimadores son insesgados o asintóticamente insesgados.\n",
    "\n",
    "$\n",
    "E(\\hat{\\theta})\n",
    "= E \\Big( \\tfrac{1}{\\bar{X}} \\Big)\n",
    "= E \\Big(\\frac{1}{\\sum_{i=1}^n \\tfrac{X_i}{n}} \\Big)\n",
    "= E \\Big(\\frac{1}{\\tfrac{1}{n} \\sum_{i=1}^n X_i} \\Big)\n",
    "= E \\Big(\\frac{n}{\\sum_{i=1}^n X_i} \\Big)\n",
    "= n E \\Big(\\frac{1}{\\sum_{i=1}^n X_i} \\Big)\n",
    "$\n",
    "\n",
    "Observemos que $Y = \\sum_{i=1}^n X_i \\sim \\Gamma(n, \\theta) \\implies E(\\hat{\\theta}) = n E \\Big(\\frac{1}{Y} \\Big)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be96ad43-2957-41af-8d30-45071b565df2",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Decidir si estos estimadores son consistentes. Justificar"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94ab44cb-ee1c-4fa8-86e4-fa2ab523f0d0",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "Se han probado 20 lámparas, obteniéndose los siguientes tiempos de duración (en días):\n",
    "\n",
    "```\n",
    "39.08 45.27 26.27 14.77 65.84 49.64 0.80 66.58 69.60 32.42 228.36 64.79 9.38 3.86 37.18 104.75 3.64 104.19 8.17 8.36\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "976d13a7-42c7-47c9-9dec-ea16712c14a8",
   "metadata": {},
   "source": [
    "Basándose en estas observaciones, estime $\\theta$ y el percentil 90 (o cuantil 0.9) del tiempo de duración de una lámpara bajo el modelo propuesto usando los estimadores hallados en el ítem (a)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "d7958f54-f81d-4480-b586-9fede1570f87",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = c(39.08, 45.27, 26.27, 14.77, 65.84, 49.64, 0.80, 66.58, 69.60, 32.42, 228.36, 64.79, 9.38, 3.86, 37.18, 104.75, 3.64, 104.19, 8.17, 8.36)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "40c9d961-402b-46be-ac59-d33ed81a390b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0203469148990284"
      ],
      "text/latex": [
       "0.0203469148990284"
      ],
      "text/markdown": [
       "0.0203469148990284"
      ],
      "text/plain": [
       "[1] 0.02034691"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = 1 / mean(X)\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "842329a1-39ee-4ae5-adcc-f74c663f2d96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "113.166300857925"
      ],
      "text/latex": [
       "113.166300857925"
      ],
      "text/markdown": [
       "113.166300857925"
      ],
      "text/plain": [
       "[1] 113.1663"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "113.166300857925"
      ],
      "text/latex": [
       "113.166300857925"
      ],
      "text/markdown": [
       "113.166300857925"
      ],
      "text/plain": [
       "[1] 113.1663"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = -log(0.1) / theta\n",
    "x\n",
    "\n",
    "# Otra forma usando las funciones de R directamente\n",
    "x = qexp(0.9, theta)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e48e696a-d309-4bca-8be6-ca5098c82717",
   "metadata": {},
   "source": [
    "¿Cómo haría para estimar el percentil solicitado si no conociera la distribución de las $X_i$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "9723dd9f-7a29-4e26-9e8e-c1cb11b64365",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "104.19"
      ],
      "text/latex": [
       "104.19"
      ],
      "text/markdown": [
       "104.19"
      ],
      "text/plain": [
       "[1] 104.19"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sort(X)[length(X) * 0.9]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c70314e6-80e7-430b-aa9a-af4ae3b8b236",
   "metadata": {},
   "source": [
    "### Pregunta E\n",
    "\n",
    "En un nuevo experimento se registra la duración de 350 lámparas. Los valores observados se encuentran en el archivo lamparas.txt. Repita el ítem (d) para este nuevo conjunto de datos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c9bb725d-7138-4c40-b8b0-91e81bb40952",
   "metadata": {},
   "outputs": [],
   "source": [
    "lamparas = scan('./datasets/p7/lamparas.txt')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "ecee2d0f-aa01-465d-b17f-4f68845c297e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0272399826909367"
      ],
      "text/latex": [
       "0.0272399826909367"
      ],
      "text/markdown": [
       "0.0272399826909367"
      ],
      "text/plain": [
       "[1] 0.02723998"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = 1 / mean(lamparas)\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "355f6e94-6c94-4804-8404-0c9f4443e7a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "84.5296092555948"
      ],
      "text/latex": [
       "84.5296092555948"
      ],
      "text/markdown": [
       "84.5296092555948"
      ],
      "text/plain": [
       "[1] 84.52961"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = qexp(0.9, theta)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dcf33f2-32d1-4394-ba6a-d5d903502a65",
   "metadata": {},
   "source": [
    "## Ejercicio 5\n",
    "\n",
    "El número de llamadas que recibe una central telefónica en un día es una variable aleatoria con distribución $\\mathcal{P}(\\theta)$. Se desea estimar la probabilidad de que en un determinado día se reciban exactamante 40 llamadas. Para ello se registrarán la cantidad de llamadas recibidas durante 48 días."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c57abd7-3d47-42bf-b36e-277b66c18f86",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "### Pregunta A\n",
    "\n",
    "Basándose en la muestra aleatoria $X_1, \\dots, X_n$, hallar el estimador de momentos y de máxima verosimilitud de $\\theta$.\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\bar{X} = E(X) \\iff \\bar{X} = \\theta \\implies \\hat{\\theta} = \\bar{X}$\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\\hat{\\theta} = \\bar{X}$ ([ver desarrollo](https://www.statology.org/mle-poisson-distribution/))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72392b0b-2fc8-4570-935b-0106fc17ccc1",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Deducir un estimador para la probabilidad de que en un determinado día se reciban 40 llamadas.\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "P(X_i = 40) = \\frac{e^{-\\theta} \\theta^{40}}{40!} = \\frac{e^{-\\bar{X}} \\bar{X}^{40}}{40!}\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b03ea31c-3490-480e-b270-45649b19e9ca",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "¿Cómo haría para estimar la probabilidad deseada si no conociera la distribución de las $X_i$?\n",
    "\n",
    "Casos favorables sobre casos totales."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75cd7a1e-ed9f-4d1c-9641-1214b0cc95e1",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "Durante 48 días se registran las siguientes cantidades:\n",
    "\n",
    "```\n",
    "40 46 46 45 42 44 50 31 41 42 50 34 62 32 46 50 39 42 44 41 47 42 41 50 34 47 38 40 44 45 35 51 38 41 39 34 48 35 40 40 43 36 40 49 45 47 34 45\n",
    "```\n",
    "\n",
    "Estimar $\\theta$. Estimar la probabilidad de que en un determinado día se reciban exactamente 40 llamadas utilizando los estimadores propuestos en los items b) y c)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "36bc4429-1ff1-479a-b7d0-279282b6a6e9",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = c(40, 46, 46, 45, 42, 44, 50, 31, 41, 42, 50, 34, 62, 32, 46, 50, 39, 42, 44, 41, 47, 42, 41, 50, 34, 47, 38, 40, 44, 45, 35, 51, 38, 41, 39, 34, 48, 35, 40, 40, 43, 36, 40, 49, 45, 47, 34, 45)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "5692996e-4a27-4417-83a5-8f272fa0d969",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "42.3958333333333"
      ],
      "text/latex": [
       "42.3958333333333"
      ],
      "text/markdown": [
       "42.3958333333333"
      ],
      "text/plain": [
       "[1] 42.39583"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = mean(X)\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "e283897d-a166-447b-bbe9-acfd070521d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0587496802022553"
      ],
      "text/latex": [
       "0.0587496802022553"
      ],
      "text/markdown": [
       "0.0587496802022553"
      ],
      "text/plain": [
       "[1] 0.05874968"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.0587496802022553"
      ],
      "text/latex": [
       "0.0587496802022553"
      ],
      "text/markdown": [
       "0.0587496802022553"
      ],
      "text/plain": [
       "[1] 0.05874968"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = 40\n",
    "(exp(-theta) * theta^x) / factorial(x)\n",
    "\n",
    "# Otra forma usando las funciones de R directamente\n",
    "dpois(x, theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ce79528-dfb4-479d-b3a2-1f7a230ac41c",
   "metadata": {},
   "source": [
    "### Pregunta E\n",
    "\n",
    "En un nuevo experimento se registra la cantidad de llamados en 200 días. Los valores observados se encuentran en el archivo llamadas.txt. Repita el item anterior para este nuevo conjunto de datos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "f7c76bb4-d160-4711-a3e0-f0d77bfede92",
   "metadata": {},
   "outputs": [],
   "source": [
    "llamadas = scan('./datasets/p7/llamadas.txt')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "2f19ba1e-98c3-4455-bee9-9de57bcef6b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "41.81"
      ],
      "text/latex": [
       "41.81"
      ],
      "text/markdown": [
       "41.81"
      ],
      "text/plain": [
       "[1] 41.81"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = mean(llamadas)\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "323faa86-d024-4679-8a06-a5b9c6c8e171",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0604935881823646"
      ],
      "text/latex": [
       "0.0604935881823646"
      ],
      "text/markdown": [
       "0.0604935881823646"
      ],
      "text/plain": [
       "[1] 0.06049359"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = 40\n",
    "dpois(x, theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80dd4913-4ec1-45b1-85d7-7489ebea2146",
   "metadata": {},
   "source": [
    "## Ejercicio 6\n",
    "\n",
    "Se sabe que la longitud de los ejes que fabrica un establecimiento siderúrgico tiene densidad:\n",
    "\n",
    "$f(x; \\theta) = \\tfrac{1}{\\theta} x^{\\tfrac{1}{\\theta} - 1} I_{(0, 1)}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2d4d8c3-3b69-44fe-ae06-aaf277c3da21",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Hallar el estimador de MV, $\\hat{\\theta}_{MV}$, y el estimador de momentos, $\\hat{\\theta}_M$, de $\\theta$.\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "L(\\theta) = f(x_1, \\dots, x_n) = \\prod_1^n f_{X_i}(x_i) = \\prod_1^n \\tfrac{1}{\\theta} x_i^{\\tfrac{1}{\\theta} - 1} = \\tfrac{1}{\\theta^n} \\prod_1^n x_i^{\\tfrac{1}{\\theta} - 1}\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ln(L(\\theta))\n",
    "&= ln \\Big( \\tfrac{1}{\\theta^n} \\prod_1^n x_i^{\\tfrac{1}{\\theta} - 1} \\Big) \\\\\n",
    "&= ln( \\tfrac{1}{\\theta^n} ) + ln \\Big( \\prod_1^n x_i^{\\tfrac{1}{\\theta} - 1} \\Big) \\\\\n",
    "&= -n ln(\\theta) + \\sum_1^n ln(x_i^{\\tfrac{1}{\\theta} - 1}) \\\\\n",
    "&= -n ln(\\theta) + \\sum_1^n (\\tfrac{1}{\\theta} - 1) ln(x_i) \\\\\n",
    "&= -n ln(\\theta) + (\\tfrac{1}{\\theta} - 1) \\sum_1^n ln(x_i)\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "$\\tfrac{\\partial}{\\partial \\theta} ln(L(\\theta)) = \\tfrac{-n}{\\theta} - \\tfrac{1}{\\theta^2} \\sum_1^n ln(x_i) = 0 \\implies \\hat{\\theta}_{MV} = - \\tfrac{1}{n} \\sum_1^n ln(x_i)$\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\bar{X} = E(X) = \\int_0^1 x f(x) dx = \\tfrac{1}{1 + \\theta} \\implies \\hat{\\theta}_{M} = \\tfrac{1}{\\bar{X}} - 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5379f5cf-b7b2-4af8-be4d-459bf7951176",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Decidir si $\\hat{\\theta}_{MV}$ es insesgado o asintóticamente insesgado.\n",
    "\n",
    "$\n",
    "E(\\hat{\\theta}_{MV})\n",
    "= E \\Big( - \\tfrac{1}{n} \\sum_1^n ln(x_i) \\Big)\n",
    "= - \\tfrac{1}{n} n E(ln(X))\n",
    "= - E(ln(X))\n",
    "= - \\int_0^1 ln(x) f_X(x) dx\n",
    "= - \\int_0^1 ln(x) \\tfrac{1}{\\theta} x^{\\tfrac{1}{\\theta} - 1} dx\n",
    "= - \\big[ -\\theta \\big]_0^1\n",
    "= \\theta\n",
    "$\n",
    "\n",
    "$\\implies E(\\hat{\\theta}_{MV})$ es insesgado."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6d63705-9017-4121-96d9-340c0a3f407d",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Estudiar la consistencia de ambos estimadores.\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\n",
    "V(\\hat{\\theta}_{MV})\n",
    "= V(- \\tfrac{1}{n} \\sum_1^n ln(x_i))\n",
    "= \\tfrac{1}{n^2} V(\\sum_1^n ln(x_i))\n",
    "= \\tfrac{1}{n^2} \\sum_1^n V(ln(X))\n",
    "= \\tfrac{n ln(X)}{n^2}\n",
    "= \\tfrac{ln(X)}{n}\n",
    "\\xrightarrow{n \\rightarrow \\infty} 0 \\implies V(\\hat{\\theta}_{MV})\n",
    "$ es consistente.\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\hat{\\theta}_{M} = \\tfrac{1}{\\bar{X}} - 1$\n",
    "\n",
    "Por la LGN: $\\hat{\\theta}_{M} = \\tfrac{1}{\\bar{X}} - 1 \\xrightarrow{n \\rightarrow \\infty} \\tfrac{1}{E(X)} - 1 = \\tfrac{1}{\\tfrac{1}{1 + \\theta}} - 1 = 1 + \\theta - 1 = \\theta$\n",
    "\n",
    "$\n",
    "P(|\\hat{\\theta}_{M} - \\theta| > \\epsilon)\n",
    "\\xrightarrow{n \\rightarrow \\infty} P(|\\theta - \\theta| > \\epsilon) = P(0 > \\epsilon) = 0 \\; \\forall \\epsilon > 0\n",
    "$\n",
    "\n",
    "$\\implies \\hat{\\theta}_{MV}$ es consistente."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32ee4c9f-66ec-4322-b324-fa16a851c672",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "Se eligen al azar 20 ejes, cuyas longitudes son:\n",
    "\n",
    "```\n",
    "0.5 0.7 0.8 0.95 0.9 0.6 0.2 0.85 0.3 0.2 0.76 0.55 0.48 0.8 0.76 0.13 0.15 0.67 0.9 0.95\n",
    "```\n",
    "\n",
    "Estimar $\\theta$ usando cada uno de los métodos propuestos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "c500f784-5bce-4069-83b4-ad3034582298",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.646090534979424"
      ],
      "text/latex": [
       "0.646090534979424"
      ],
      "text/markdown": [
       "0.646090534979424"
      ],
      "text/plain": [
       "[1] 0.6460905"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.656205413846233"
      ],
      "text/latex": [
       "0.656205413846233"
      ],
      "text/markdown": [
       "0.656205413846233"
      ],
      "text/plain": [
       "[1] 0.6562054"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta_M = function(X) 1 / mean(X) - 1\n",
    "theta_MV = function(X) -sum(log(X)) / length(X)\n",
    "\n",
    "X = c(0.5, 0.7, 0.8, 0.95, 0.9, 0.6, 0.2, 0.85, 0.3, 0.2, 0.76, 0.55, 0.48, 0.8, 0.76, 0.13, 0.15, 0.67, 0.9, 0.95)\n",
    "theta_M(X)\n",
    "theta_MV(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d88064b-52be-4bd7-bd94-df84a3f2f820",
   "metadata": {},
   "source": [
    "## Ejercicio 7\n",
    "\n",
    "Un estado tiene varios distritos. Supongamos que cada distrito tiene igual proporción $\\theta$ de personas que están a favor de una propuesta de control de armas. En cada uno de 8 distritos elegidos al azar, se cuenta la cantidad de personas que hay que encuestar hasta encontrar alguna de acuerdo con la propuesta (llamemos $X$ a esta variable aleatoria). Los resultados son: 3, 8, 9, 6, 4, 5, 3, 2 (i.e.: en el primer distrito las dos primeras personas encuestadas estaban en contra y la tercera a favor)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "c0f2bce9-47d4-4379-8360-dca278fa737c",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = c(3, 8, 9, 6, 4, 5, 3, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d4b4a7c-fd41-4e71-ae34-cbaadbb905de",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "¿Qué distribución tiene $X$?\n",
    "\n",
    "Geométrica: \"cantidad de repeticiones hasta el primer éxito\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0af270d-9707-4972-a1f4-9531baf350d7",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Estimar $\\theta$ por el método de máxima verosimiltud y por el método de los momentos.\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\\hat{\\theta}_{MV} = \\tfrac{1}{\\bar{X}}$ ([ver desarrollo](https://www.projectrhea.org/rhea/index.php/MLE_Examples:_Exponential_and_Geometric_Distributions_Old_Kiwi))\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\bar{X} = E(X) = \\tfrac{1}{\\theta} \\implies \\hat{\\theta}_{M} = \\tfrac{1}{\\bar{X}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8588b31-6d30-4ef3-b4ee-69f8b271e6ee",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Estimar por máxima verosimilitud $P_\\theta(X \\geq 5)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "ef1ef729-91cf-4ba6-b935-f8443de644a3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.4096"
      ],
      "text/latex": [
       "0.4096"
      ],
      "text/markdown": [
       "0.4096"
      ],
      "text/plain": [
       "[1] 0.4096"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = 5\n",
    "theta = 1 / mean(X)\n",
    "pgeom(x - 2, theta, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6bfbfa10-bfd0-41be-b290-d62fab9b3e43",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "Estimar $P_\\theta(X \\geq 5)$ sin asumir que $X$ tiene distribución geométrica. Comparar el resultado con el del ítem anterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "0ae9ef76-8f31-4adb-8629-baf0c6f28383",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.5"
      ],
      "text/latex": [
       "0.5"
      ],
      "text/markdown": [
       "0.5"
      ],
      "text/plain": [
       "[1] 0.5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum(X >= 5) / length(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf22b0f9-6e1c-4b92-84cd-444d83cc5026",
   "metadata": {},
   "source": [
    "## Ejercicio 8\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria de una distribución $\\mathcal{U}(0, \\theta)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b0b320d-7c5f-4ca0-80bf-cd24d68ffd74",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "### Pregunta A\n",
    "\n",
    "Probar que $M_n = max_{1 \\leq i \\leq n} X_i$ es el estimador de máxima verosimilitud de $\\theta$.\n",
    "\n",
    "Intuitivamente podemos ver que si $X$ tiene distribución $\\mathcal{U}(0, \\theta)$, necesitamos un $\\theta$ más grande que todas las $X_i$ observadas, pues sino, habría una observación que cae en el intervalo $(\\theta, +\\infty)$ que técnicamente debería tenerla probabilidad 0, y eso sería absurdo.\n",
    "\n",
    "Entonces el mejor estimador que tenemos es asignar $\\theta$ al $X_i$ más grande. Notar que el verdadero $\\theta$ podría ser aún más grande, pero no tenemos más información para justificar eso y además si tomamos un $\\theta$ más grande que el máximo, ya no maximizamos la función de máxima verosimilitud.\n",
    "\n",
    "*Nota: La demostración formal está desarrollada en el apunte de Bianco y Martínez (2004), página 168.*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84aaa801-df8d-470f-9e8a-cfe17c63c972",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Calcular el estimador de $\\theta$ basado en el primer momento.\n",
    "\n",
    "$\\bar{X} = E(X) = \\tfrac{\\theta}{2} \\implies \\hat{\\theta}_M = 2\\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "371075cc-85f2-481e-bfc3-7de2e5e3d6c2",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Decir si los estimadores obtenidos son insesgados o asintóticamente insesgados, y consistentes. Justificar.\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "Recordemos que los estimadores son en sí mismos variables aleatorias. Entonces para calcular la esperanza de $\\hat{\\theta}_{MV}$ vamos a necesitar conocer su función de densidad. Y para eso primero buscamos la función acumulada y luego la derivamos.\n",
    "\n",
    "Sea $U = max_{1 \\leq i \\leq n} X_i$ una v.a. (podríamos llamarla $\\hat{\\theta}_{MV}$ también pero así quedá más claro para leer).\n",
    "\n",
    "\\begin{align*}\n",
    "F_U(u)\n",
    "= P(U \\leq u)\n",
    "= P(max_{1 \\leq i \\leq n} X_i \\leq u)\n",
    "\\end{align*}\n",
    "\n",
    "Observación: $max_{1 \\leq i \\leq n} X_i \\leq u \\implies X_i \\leq u \\; \\forall i$\n",
    "\n",
    "Es decir, si el $X_i$ máximo es menor que $u$, entonces todos los $X_i$ son menores que $u$:\n",
    "\n",
    "\\begin{align*}\n",
    "P(max_{1 \\leq i \\leq n} X_i \\leq u)\n",
    "= P(\\cap_{i=1}^n X_i \\leq u)\n",
    "= \\prod_{i=1}^n P(X_i \\leq u)\n",
    "= \\prod_{i=1}^n F_{X}(u)\n",
    "= F_{X}(u)^n\n",
    "\\end{align*}\n",
    "\n",
    "Para el desarrollo anterior usamos que todas las $X_i$ son i.i.d. Recapitulando:\n",
    "\n",
    "\\begin{align*}\n",
    "F_U(u) = F_{X}(u)^n\n",
    "\\end{align*}\n",
    "\n",
    "Como $X \\sim \\mathcal{U}(0, \\theta)$, sabemos exactamente cómo es $F_X$:\n",
    "\n",
    "\\begin{align*}\n",
    "F_U(u) = \\begin{cases}\n",
    "0 & u < 0 \\\\\n",
    "\\big( \\tfrac{u}{\\theta} \\big) ^n & 0 \\leq u \\leq \\theta \\\\\n",
    "1 & \\theta < u\n",
    "\\end{cases}\n",
    "\\end{align*}\n",
    "\n",
    "Muy bien, ahora que tenemos $F_U$ podemos derivar para obtener $f_u$:\n",
    "\n",
    "\\begin{align*}\n",
    "f_U(u) = \\tfrac{\\partial}{\\partial u} F_U(u) = \\tfrac{n u^{n-1}}{\\theta^n} I_{[0, \\theta]}(u)\n",
    "\\end{align*}\n",
    "\n",
    "Finalmente podemos calcular la esperanza de $\\hat{\\theta}_{MV}$:\n",
    "\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta}_{MV})\n",
    "= E(U)\n",
    "= \\int_0^\\theta u f_U(u) du\n",
    "= \\int_0^\\theta u \\tfrac{n u^{n-1}}{\\theta^n} du\n",
    "= \\tfrac{n}{\\theta^n} \\int_0^\\theta u^n du\n",
    "= \\tfrac{n}{\\theta^n} \\Big[ \\tfrac{u^{n+1}}{n+1} \\big]_0^\\theta\n",
    "= \\tfrac{n}{\\theta^n} * \\tfrac{\\theta^{n+1}}{n+1}\n",
    "= \\tfrac{n}{n+1} \\theta \\\\\n",
    "\\implies E(\\hat{\\theta}_{MV}) = \\tfrac{n}{n+1} \\theta\n",
    "\\end{align*}\n",
    "\n",
    "$\\hat{\\theta}_{MV}$ es asintóticamente insesgado pues:\n",
    "\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta}_{MV}) = \\tfrac{n}{n+1} \\theta \\xrightarrow{n \\rightarrow \\infty} \\theta\n",
    "\\end{align*}\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$E(\\hat{\\theta}_M) = E(2\\bar{X}) = 2 E(\\bar{X}) = 2 E(X_1) = 2 \\tfrac{\\theta}{2} = \\theta$\n",
    "\n",
    "$\\implies \\hat{\\theta}_M$ es insesgado pues su esperanza coincide con el parámetro que estima.\n",
    "\n",
    "**Consistencia**\n",
    "\n",
    "*Ver el apunte de Bianco y Martínez (2004), página 176.*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ec2999d2-fa47-4ef7-aa12-e68859aaaa4d",
   "metadata": {},
   "source": [
    "## Ejercicio 9\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria de una distribución Rayleigh, cuya densidad está dada por:\n",
    "\n",
    "$f(x; \\theta) = \\tfrac{x}{\\theta} e^{-\\tfrac{x^2}{2\\theta}} I_{[0, \\infty)}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf4b2bb0-2134-484c-a8dd-6d5f6cf6fa9a",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Hallar el estimador de máxima verosimilitud de $\\theta$.\n",
    "\n",
    "$\n",
    "L(\\theta)\n",
    "= f(x_1, \\dots, x_n)\n",
    "= \\prod_{i=1}^n f_{X_i}(x_i)\n",
    "= \\prod_{i=1}^n \\frac{x_i}{\\theta} e^{-\\tfrac{x_i^2}{2\\theta}} I_{[0, \\infty)}(x_i)\n",
    "= \\frac{1}{\\theta^n} \\Big( \\prod_{i=1}^n x_i \\Big) \\Big( e^{-\\tfrac{1}{2\\theta} \\sum_{i=1}^n x_i^2} \\Big) \\Big( \\prod_{i=1}^n I_{[0, \\infty)}(x_i) \\Big)\n",
    "$\n",
    "\n",
    "$ln(L(\\theta)) = -n ln(\\theta) + \\sum_{i=1}^n ln(x_i) - \\frac{1}{2\\theta} \\sum_{i=1}^n x_i^2$\n",
    "\n",
    "$\\tfrac{\\partial}{\\partial \\theta} ln(L(\\theta)) = \\tfrac{-n}{\\theta} + \\frac{1}{2\\theta^2} \\sum_{i=1}^n x_i^2 = 0 \\iff \\hat{\\theta}_{MV} = \\tfrac{1}{2n} \\sum_{i=1}^n x_i^2 = \\tfrac{\\overline{X^2}}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7625ec18-48b4-4872-8b1a-109c0266586e",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Decir si el estimador obtenido es insesgado o asintóticamente insesgado, y consistente. Justificar.\n",
    "\n",
    "(Sugerencia: Hallar la distribución de la v.a. $X_1^2$).\n",
    "\n",
    "$Y = X^2$\n",
    "\n",
    "$F_Y(y) = P(Y \\leq y) = P(X^2 \\leq y) = P(|X| \\leq \\sqrt{y}) = P(X \\leq \\sqrt{y}) - P(X \\leq -\\sqrt{y}) = F_X(\\sqrt{y}) - F_X(-\\sqrt{y})$\n",
    "\n",
    "Observación: $I_{[0, \\infty)}(x) \\implies F_X(-\\sqrt{y}) = 0$\n",
    "\n",
    "$F_Y(y) = F_X(\\sqrt{y})$\n",
    "\n",
    "$\n",
    "f_Y(y)\n",
    "= f_X(\\sqrt{y}) \\tfrac{1}{2\\sqrt{y}}\n",
    "= \\tfrac{\\sqrt{y}}{\\theta} e^{-\\tfrac{\\sqrt{y}^2}{2\\theta}} \\tfrac{1}{2\\sqrt{y}} I_{[0, \\infty)}(\\sqrt{y})\n",
    "= \\tfrac{1}{2\\theta} e^{-\\tfrac{y}{2\\theta}} I_{[0, \\infty)}(y)\n",
    "$\n",
    "\n",
    "$Y = X^2 \\sim \\mathcal{E}(\\tfrac{1}{2\\theta})$\n",
    "\n",
    "$E(X^2) = 2\\theta$\n",
    "\n",
    "Ahora veamos cuál es la esperanza del estimador:\n",
    "\n",
    "$\n",
    "E(\\hat{\\theta}_{MV})\n",
    "= E(\\tfrac{1}{2n} \\sum_{i=1}^n x_i^2)\n",
    "= \\tfrac{1}{2n} \\sum_{i=1}^n E(x_i^2)\n",
    "= \\tfrac{1}{2n} \\sum_{i=1}^n E(X^2)\n",
    "= \\tfrac{1}{2n} \\sum_{i=1}^n 2\\theta\n",
    "= \\tfrac{1}{2n} 2n \\theta = \\theta\n",
    "$\n",
    "\n",
    "$\\implies \\hat{\\theta}_{MV}$ es insesgado.\n",
    "\n",
    "Veamos la consistencia.\n",
    "\n",
    "Por la LGN: $\\hat{\\theta}_{MV} = \\tfrac{\\overline{X^2}}{2} \\xrightarrow{n \\rightarrow \\infty} \\tfrac{E(X^2)}{2} = \\tfrac{2\\theta}{2} = \\theta$\n",
    "\n",
    "$\n",
    "P(|\\hat{\\theta}_{MV} - \\theta| > \\epsilon)\n",
    "\\xrightarrow{n \\rightarrow \\infty} P(|\\theta - \\theta| > \\epsilon) = P(0 > \\epsilon) = 0 \\; \\forall \\epsilon > 0\n",
    "$\n",
    "\n",
    "$\\implies \\hat{\\theta}_{MV}$ es consistente."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4fe27ea8-a997-4e5b-b751-9f6548750ef3",
   "metadata": {},
   "source": [
    "## Ejercicio 10\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria de una distribución con densidad:\n",
    "\n",
    "$f(x; \\theta) = e^{-(x - \\theta)} I_{[\\theta, \\infty)}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "176e5529-d18c-4f24-ad81-3db09f7e82ef",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Probar que $m_n = min_{1 \\leq i \\leq n} X_i$ es el estimador de máxima verosimilitud de $\\theta$.\n",
    "\n",
    "Planteamos la función de máxima verosimilitud:\n",
    "\n",
    "\\begin{align*}\n",
    "L(\\theta) &= f(x_1, \\dots, x_n) = \\prod_{i=1}^n f(x_i) \\\\\n",
    "&= \\prod_{i=1}^n e^{-(x_i - \\theta)} I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "&= \\prod_{i=1}^n e^{-(x_i - \\theta)} \\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "&= e^{\\sum_{i=1}^n -(x_i - \\theta)} \\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "&= e^{\\sum_{i=1}^n (-x_i + \\theta)} \\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "&= e^{\\sum_{i=1}^n -x_i} e^{\\sum_{i=1}^n \\theta} \\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "&= e^{\\sum_{i=1}^n -x_i} e^{n \\theta} \\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "Ahora queremos ver qué valores de $\\theta$ maximizan $L(\\theta)$. Analizamos cada término:\n",
    "\n",
    "$e^{\\sum_{i=1}^n -x_i}$ Es una constante, así que podemos ignorar este término.\n",
    "\n",
    "\n",
    "$e^{n\\theta} \\xrightarrow{\\theta \\rightarrow \\infty} \\infty$ Esto nos dice que queremos el $\\theta$ más grande posible.\n",
    "\n",
    "Pero también necesitamos que la productoria de la indicadora de 1 para todos los $X_i$. Veamos cuándo pasa eso:\n",
    "\n",
    "\\begin{align*}\n",
    "\\prod_{i=1}^n I_{\\{x_i \\geq \\theta\\}} = 1 \\iff \\theta \\leq X_i \\; \\forall i, 1 \\leq i \\leq n \\implies \\theta \\leq min_{1 \\leq i \\leq n} X_i\n",
    "\\end{align*}\n",
    "\n",
    "Por lo tanto el $\\theta$ más grande que podemos elegir es el mínimo de todos los $X_i$ y ese es el estimador de máxima verosimilitud:\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{\\theta} = min_{1 \\leq i \\leq n} X_i\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e899cfac-9975-479d-ac3b-4bc00323d2b1",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Calcular el estimador de $\\theta$ basado en el primer momento.\n",
    "\n",
    "$\\bar{X} = E(X) = \\int_\\theta^\\infty e^{-(x - \\theta)} = \\theta + 1 \\implies \\hat{\\theta}_M = \\bar{X} - 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18e6f9a3-b17c-4ffd-891a-d0d7878017b7",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Decir si los estimadores obtenidos son insesgados o asintóticamente insesgados, y consistentes. Justificar.\n",
    "\n",
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "*Nota: Este desarrollo es análogo al del ejercicio 8c.*\n",
    "\n",
    "Para ver la consistencia, veamos primero el sesgo del estimador y luego buscamos su varianza.\n",
    "\n",
    "Sea $U = min_{1 \\leq i \\leq n} X_i$\n",
    "\n",
    "\\begin{align*}\n",
    "F_U(u) = P(U \\leq u) = 1 - P(U \\geq u) = 1 - P(min_{1 \\leq i \\leq n} X_i \\geq u)\n",
    "\\end{align*}\n",
    "\n",
    "Observación: $min_{1 \\leq i \\leq n} X_i \\geq u \\implies X_i \\geq u \\; \\forall i$\n",
    "\n",
    "Es decir, si el $X_i$ mínimo es mayor que $u$, entonces todos los $X_i$ son mayores que $u$.\n",
    "\n",
    "\\begin{align*}\n",
    "1 - P(min_{1 \\leq i \\leq n} \\geq u)\n",
    "&= 1 - P(\\cap_{i=1}^n X_i \\geq u) \\\\\n",
    "&= 1 - \\prod_{i=1}^n P(X_i \\geq u) \\\\\n",
    "&= 1 - \\prod_{i=1}^n (1 - P(X_i \\leq u)) \\\\\n",
    "&= 1 - \\prod_{i=1}^n (1 - F_{X_i}(u)) \\\\\n",
    "&= 1 - \\prod_{i=1}^n (1 - F_X(u)) \\\\\n",
    "&= 1 - (1 - F_X(u))^n\n",
    "\\end{align*}\n",
    "\n",
    "Para el desarrollo anterior usamos que todas las $X_i$ son i.i.d. Recapitulando:\n",
    "\n",
    "\\begin{align*}\n",
    "F_U(u) = 1 - (1 - F_X(u))^n\n",
    "\\end{align*}\n",
    "\n",
    "No sabemos cómo es $F_X$, calculemos eso antes de seguir:\n",
    "\n",
    "\\begin{align*}\n",
    "F_X(t) = \\int_\\theta^t f(x) dx = \\int_\\theta^t e^{-(x - \\theta)} dx = 1 - e^{\\theta - t}\n",
    "\\end{align*}\n",
    "\n",
    "Ahora sí podemos seguir. Derivamos ambos lados de la igualdad para obtener $f_U$:\n",
    "\n",
    "\\begin{align*}\n",
    "\\tfrac{\\partial}{\\partial u} F_U(u) = f_U(u) = n (1 - F_X(u))^{n-1} f_X(u) = n (e^{\\theta - u})^{n-1} e^{-(u - \\theta)} = ne^{n(\\theta - u)}\n",
    "\\end{align*}\n",
    "\n",
    "Finalmente podemos calcular la esperanza de $\\hat{\\theta}$:\n",
    "\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta}) = E(U) = \\int_\\theta^\\infty u f_U(u) du = \\int_\\theta^\\infty u ne^{n(\\theta - u)} du = \\theta + \\frac{1}{n}\n",
    "\\end{align*}\n",
    "\n",
    "Veamos que $\\hat{\\theta}$ es asintóticamente insesgado pues:\n",
    "\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta}) = \\theta + \\frac{1}{n} \\xrightarrow{n \\rightarrow \\infty} \\theta\n",
    "\\end{align*}\n",
    "\n",
    "Calculemos $E(\\hat{\\theta}^2)$ para poder luego calcular la varianza:\n",
    "\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta}^2) = E(U^2) = \\int_\\theta^\\infty u^2 f_U(u) du = \\int_\\theta^\\infty u^2 ne^{n(\\theta - u)} du = \\frac{\\theta^2 n^2 + 2 \\theta n + 2}{n^2}\n",
    "\\end{align*}\n",
    "\n",
    "Ahora sí, calculemos la varianza de $\\hat{\\theta}$ y veamos si es consistente:\n",
    "\n",
    "\\begin{align*}\n",
    "V(\\hat{\\theta}) = E(\\hat{\\theta}^2) - E(\\hat{\\theta})^2 &= \\frac{\\theta^2 n^2 + 2 \\theta n + 2}{n^2} - \\Big( \\theta + \\frac{1}{n} \\Big)^2 \\\\\n",
    "&= \\theta^2 + \\frac{2 \\theta}{n} + \\frac{2}{n^2} - \\theta^2 - \\frac{2 \\theta}{n} - \\frac{1}{n^2} \\\\\n",
    "&= \\frac{1}{n^2} \\xrightarrow{n \\rightarrow \\infty} 0\n",
    "\\end{align*}\n",
    "\n",
    "Por lo tanto, como se cumplen estas 2 condiciones:\n",
    "\n",
    "\\begin{align*}\n",
    "\\lim_{n \\rightarrow \\infty} E(\\hat{\\theta}) = \\theta \\land \\lim_{n \\rightarrow \\infty} V(\\hat{\\theta}) = 0 \\implies \\hat{\\theta} \\; \\text{es consistente}\n",
    "\\end{align*}\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$E(\\hat{\\theta}_M)\n",
    "= E(\\bar{X} - 1)\n",
    "= E(\\bar{X}) - 1\n",
    "= E(\\tfrac{1}{n} \\sum_{i=1}^n X_i) - 1\n",
    "= \\Big( \\tfrac{1}{n} \\sum_{i=1}^n E(X_i) \\Big) - 1\n",
    "= \\Big( \\tfrac{1}{n} \\sum_{i=1}^n E(X) \\Big) - 1\n",
    "= \\tfrac{1}{n} n E(X) - 1\n",
    "= E(X) - 1\n",
    "= \\theta + 1 - 1\n",
    "= \\theta\n",
    "$\n",
    "\n",
    "$\\implies \\hat{\\theta}_M$ es insesgado pues su esperanza coincide con el parámetro que estima.\n",
    "\n",
    "Veamos la consistencia.\n",
    "\n",
    "Por la LGN: $\\bar{X} \\xrightarrow{n \\rightarrow \\infty} E(X) = \\theta + 1$\n",
    "\n",
    "$\n",
    "P(|\\hat{\\theta}_M - \\theta| > \\epsilon)\n",
    "= P(|\\bar{X} - 1 - \\theta| > \\epsilon)\n",
    "\\xrightarrow{n \\rightarrow \\infty} P(|\\theta + 1 - 1 - \\theta| > \\epsilon)\n",
    "= P(0 > \\epsilon) = 0 \\; \\forall \\epsilon > 0\n",
    "$\n",
    "\n",
    "$\\implies \\hat{\\theta}_M$ es consistente."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6e6a7d3-b1f8-4486-9995-98b008ac7c2d",
   "metadata": {},
   "source": [
    "## Ejercicio 11\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria de una distribución con media $\\mu$ y varianza $\\sigma^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfc021a0-f0ec-4506-b696-0dacf37c6553",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "### Pregunta A\n",
    "\n",
    "Probar que $\\bar{X}^2$ no es un estimador insesgado de $\\mu^2$. ¿Es asintóticamente insesgado? ¿Es consistente?\n",
    "\n",
    "El truco acá es depejar $E(\\bar{X}^2)$ usando la fórmula de la varianza: $V(\\bar{X}) = E(\\bar{X}^2) - E(\\bar{X})^2$ y los recursos ya usados extensamente para despejar la esperanza y varianza de $\\bar{X}$ en función de $X$ (suponiendo que todas las $X_i$ son i.i.d.).\n",
    "\n",
    "$\n",
    "E(\\bar{X}^2)\n",
    "= V(\\bar{X}) + E(\\bar{X})^2\n",
    "= \\tfrac{V(X)}{n} + E(X)^2\n",
    "= \\tfrac{\\sigma^2}{n} + \\mu^2\n",
    "\\neq \\mu^2\n",
    "$\n",
    "\n",
    "Es asintóticamente insesgado pues $E(\\bar{X}^2) = \\tfrac{\\sigma^2}{n} + \\mu^2 \\xrightarrow{n \\rightarrow \\infty} \\mu^2$\n",
    "\n",
    "Por la LGN: $\\bar{X}^2 \\xrightarrow{n \\rightarrow \\infty} E(X)^2$\n",
    "\n",
    "$\n",
    "P(|\\bar{X}^2 - \\mu^2| > \\epsilon)\n",
    "\\xrightarrow{n \\rightarrow \\infty} P(|E(X)^2 - \\mu^2| > \\epsilon)\n",
    "= P(|\\mu^2 - \\mu^2| > \\epsilon)\n",
    "= P(0 > \\epsilon) = 0 \\; \\forall \\epsilon > 0\n",
    "$\n",
    "\n",
    "$\\implies \\bar{X}^2$ es consistente."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9e9b208-acdc-4375-828f-13e37ac2a555",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿Para qué valores de $k$ es $\\hat{\\mu}^2 = (\\bar{X}^2 - ks^2)$ un estimador insesgado de $\\mu^2$?\n",
    "\n",
    "$E(\\hat{\\mu}^2) = E(\\bar{X}^2 - ks^2) = E(\\bar{X}^2) - kE(s^2) = \\tfrac{\\sigma^2}{n} + \\mu^2 - kE(s^2)$\n",
    "\n",
    "Observación: como $s^2$ es un estimador insesgado de $\\sigma^2 \\implies E(s^2) = \\sigma^2$\n",
    "\n",
    "$E(\\hat{\\mu}^2) = \\tfrac{\\sigma^2}{n} + \\mu^2 - k \\sigma^2 = \\mu^2 \\iff k = \\tfrac{1}{n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "119bb515-99e2-4338-b395-68b8d8d6c678",
   "metadata": {},
   "source": [
    "## Ejercicio 12\n",
    "\n",
    "La cantidad de días de internación de los pacientes que llegan a la guardia de un hospital es una variable aleatoria $Y$ con distribución binomial negativa con media $\\mu$ y parámetro de dispersión $r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "4f1d19a4-b058-488e-89a6-222539f7c3b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      size          mu    \n",
       "  1.00067475   4.79208832 \n",
       " (0.05693679) (0.16655823)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(MASS)\n",
    "dias = scan('./datasets/p7/dias.txt')\n",
    "fitdistr(dias, densfun='negative binomial')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "edcaba6a-3f9d-4a0a-8448-73d4f3095f4a",
   "metadata": {},
   "source": [
    "## Ejercicio 13\n",
    "\n",
    "Se define el error cuadrático medio de un estimador $\\hat{\\theta}$ como:\n",
    "\n",
    "$ECM(\\hat{\\theta}) = E[(\\hat{\\theta} - \\theta)^2]$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fef7d4fc-9bb9-432d-9cde-92bcdb15a99e",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Verificar que $ECM(\\hat{\\theta}) = V(\\hat{\\theta}) + \\text{sesgo}(\\hat{\\theta})^2$ donde $\\text{sesgo}(\\hat{\\theta}) = E(\\hat{\\theta}) - \\theta$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ECM(\\hat{\\theta})\n",
    "&= V(\\hat{\\theta}) + \\text{sesgo}(\\hat{\\theta})^2 \\\\\n",
    "&= E(\\hat{\\theta}^2) - E(\\hat{\\theta})^2 + (E(\\hat{\\theta}) - \\theta)^2 \\\\\n",
    "&= E(\\hat{\\theta}^2) - E(\\hat{\\theta})^2 + E(\\hat{\\theta})^2 - 2 \\theta E(\\hat{\\theta}) + \\theta^2 \\\\\n",
    "&= E(\\hat{\\theta}^2 - 2 \\theta \\hat{\\theta} + \\theta^2) \\\\\n",
    "&= E((\\hat{\\theta} - \\theta)^2)\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ddfa1167-e553-4bdf-9a86-625d18f14fbd",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿Cuánto vale $ECM(\\hat{\\theta})$ si $\\hat{\\theta}$ es un estimador insesgado de $\\theta$?\n",
    "\n",
    "Vamos a utilizar la propiedad de que la esperanza de un estimador insesgado es el valor real del parámetro estimado (es decir $E(\\hat{\\theta}) = \\theta$ si $\\hat{\\theta}$ es insesgado).\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ECM(\\hat{\\theta})\n",
    "&= V(\\hat{\\theta}) + \\text{sesgo}(\\hat{\\theta})^2 \\\\\n",
    "&= V(\\hat{\\theta}) + (E(\\hat{\\theta}) - \\theta)^2 \\\\\n",
    "&= V(\\hat{\\theta}) + (\\theta - \\theta)^2 \\\\\n",
    "&= V(\\hat{\\theta})\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e6c64db-4df2-45f0-b675-932f618ea652",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Consideremos un estimador de la varianza de la forma $\\hat{\\sigma}^2 = ks^2$, siendo $s^2$ la varianza muestral. Hallar el valor de $k$ que minimiza $ECM(\\hat{\\sigma}^2)$.\n",
    "\n",
    "Sugerencia: Usar que $E(s^4) = \\tfrac{n+1}{n-1}\\sigma^4$\n",
    "\n",
    "Y recordemos que: $E(s^2) = \\sigma^2$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ECM(\\hat{\\sigma}^2)\n",
    "&= ECM(ks^2) \\\\\n",
    "&= V(ks^2) + (E(ks^2) - \\sigma^2)^2 \\\\\n",
    "&= E[(ks^2)^2] - E(ks^2)^2 + (kE(s^2) - \\sigma^2)^2 \\\\\n",
    "&= k^2E(s^4) - k^2E(s^2)^2 + k^2E(s^2)^2 - 2\\sigma^2kE(s^2) + \\sigma^4 \\\\\n",
    "&= k^2E(s^4) - 2\\sigma^2kE(s^2) + \\sigma^4 \\\\\n",
    "&= \\tfrac{n+1}{n-1}\\sigma^4k^2 - 2\\sigma^4k + \\sigma^4 \\\\\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "Tenemos un polinomio de grado 2 con coeficiente principal positivo, por lo tanto su gráfico es una parábola \"feliz\", y el $k$ mínimo corresponde al vértice de la parábola.\n",
    "\n",
    "**Opción 1: buscamos el vértice de la parábola usando la fórmula**\n",
    "\n",
    "$\n",
    "k_{\\text{min}}\n",
    "= \\frac{- (- 2\\sigma^4)}{2 \\tfrac{n+1}{n-1}\\sigma^4}\n",
    "= \\frac{2\\sigma^4}{2\\sigma^4\\tfrac{n+1}{n-1}}\n",
    "= \\tfrac{n-1}{n+1}\n",
    "$\n",
    "\n",
    "**Opción 2: buscamos dónde se hace cero la derivada**\n",
    "\n",
    "$\n",
    "\\tfrac{\\partial}{\\partial k} ECM(\\hat{\\sigma}^2) = \\tfrac{n+1}{n-1}\\sigma^4 2k - 2\\sigma^4 = 0 \\iff k_{\\text{min}} = \\tfrac{n-1}{n+1}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aacc9b96-ab7f-46fe-98a3-a9a38efcb9b7",
   "metadata": {},
   "source": [
    "## Ejercicio 14\n",
    "\n",
    "En el Ejercicio 8, calcular el ECM de los estimadores calculados en (a) y (b) y compararlos. En función de esta comparación, ¿cuál de los dos estimadores usaría?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "931554bd-48aa-4c09-a456-69667050ac97",
   "metadata": {},
   "source": [
    "**Estimador de máxima verosimilitud**\n",
    "\n",
    "$\\hat{\\theta}_{MV} = max_{1 \\leq i \\leq n} X_i$\n",
    "\n",
    "$E(\\hat{\\theta}_{MV}) = \\tfrac{n}{n+1} \\theta$\n",
    "\n",
    "$V(\\hat{\\theta}_{MV}) = \\frac{n}{(n+2)(n+1)^2} \\theta^2$\n",
    "\n",
    "*Nota: Ver el cálculo de esta varianza en el apunte de Bianco y Martínez (2004), página 176.*\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ECM(\\hat{\\theta}_{MV})\n",
    "&= V(\\hat{\\theta}_{MV}) + (E(\\hat{\\theta}_{MV}) - \\theta)^2 \\\\\n",
    "&= \\frac{n}{(n+2)(n+1)^2} \\theta^2 + (\\tfrac{n}{n+1} \\theta - \\theta)^2 \\\\\n",
    "&= \\frac{2}{(n+2)(n+1)} \\theta^2\n",
    "\\end{align*}\n",
    "$\n",
    "[ver desarrollo](https://www.wolframalpha.com/input/?i=n%2F%28%28n%2B2%29%28n%2B1%29%5E2%29+%5Ctheta%5E2+%2B+%28n%2F%28n%2B1%29+%5Ctheta+-+%5Ctheta%29%5E2)\n",
    "\n",
    "**Estimador de momentos**\n",
    "\n",
    "$\\hat{\\theta}_M = 2\\bar{X}$\n",
    "\n",
    "$E(\\hat{\\theta}_M) = \\theta$\n",
    "\n",
    "$\n",
    "V(\\hat{\\theta}_M)\n",
    "= V(2\\bar{X})\n",
    "= 4V(\\bar{X})\n",
    "= \\tfrac{4}{n} V(X)\n",
    "= \\tfrac{4}{n} \\tfrac{\\theta^2}{12}\n",
    "= \\tfrac{\\theta^2}{3n}\n",
    "$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "ECM(\\hat{\\theta}_M)\n",
    "&= V(\\hat{\\theta}_M) + (E(\\hat{\\theta}_M) - \\theta)^2 \\\\\n",
    "&= \\tfrac{\\theta^2}{3n} + (\\theta - \\theta)^2 \\\\\n",
    "&= \\tfrac{\\theta^2}{3n}\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "**Conclusión**\n",
    "\n",
    "El ECM de $\\hat{\\theta}_{MV}$ tiene un $n^2$ en el denominador, mientras que el ECM de $\\hat{\\theta}_M$ tiene solo un $n$ en el denominador. Por lo tanto usaría $\\hat{\\theta}_{MV}$ ya que el error va a ser menor a medida que $n \\rightarrow \\infty$, pues una función cuadrática crece más rápido que una lineal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "8568467d-645d-49a6-abfd-a7dd91e87036",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAYAAAD958/bAAAEDmlDQ1BrQ0dDb2xvclNwYWNl\nR2VuZXJpY1JHQgAAOI2NVV1oHFUUPpu5syskzoPUpqaSDv41lLRsUtGE2uj+ZbNt3CyTbLRB\nkMns3Z1pJjPj/KRpKT4UQRDBqOCT4P9bwSchaqvtiy2itFCiBIMo+ND6R6HSFwnruTOzu5O4\na73L3PnmnO9+595z7t4LkLgsW5beJQIsGq4t5dPis8fmxMQ6dMF90A190C0rjpUqlSYBG+PC\nv9rt7yDG3tf2t/f/Z+uuUEcBiN2F2Kw4yiLiZQD+FcWyXYAEQfvICddi+AnEO2ycIOISw7UA\nVxieD/Cyz5mRMohfRSwoqoz+xNuIB+cj9loEB3Pw2448NaitKSLLRck2q5pOI9O9g/t/tkXd\na8Tbg0+PszB9FN8DuPaXKnKW4YcQn1Xk3HSIry5ps8UQ/2W5aQnxIwBdu7yFcgrxPsRjVXu8\nHOh0qao30cArp9SZZxDfg3h1wTzKxu5E/LUxX5wKdX5SnAzmDx4A4OIqLbB69yMesE1pKojL\njVdoNsfyiPi45hZmAn3uLWdpOtfQOaVmikEs7ovj8hFWpz7EV6mel0L9Xy23FMYlPYZenAx0\nyDB1/PX6dledmQjikjkXCxqMJS9WtfFCyH9XtSekEF+2dH+P4tzITduTygGfv58a5VCTH5Pt\nXD7EFZiNyUDBhHnsFTBgE0SQIA9pfFtgo6cKGuhooeilaKH41eDs38Ip+f4At1Rq/sjr6NEw\nQqb/I/DQqsLvaFUjvAx+eWirddAJZnAj1DFJL0mSg/gcIpPkMBkhoyCSJ8lTZIxk0TpKDjXH\nliJzZPO50dR5ASNSnzeLvIvod0HG/mdkmOC0z8VKnzcQ2M/Yz2vKldduXjp9bleLu0ZWn7vW\nc+l0JGcaai10yNrUnXLP/8Jf59ewX+c3Wgz+B34Df+vbVrc16zTMVgp9um9bxEfzPU5kPqUt\nVWxhs6OiWTVW+gIfywB9uXi7CGcGW/zk98k/kmvJ95IfJn/j3uQ+4c5zn3Kfcd+AyF3gLnJf\ncl9xH3OfR2rUee80a+6vo7EK5mmXUdyfQlrYLTwoZIU9wsPCZEtP6BWGhAlhL3p2N6sTjRdd\nuwbHsG9kq32sgBepc+xurLPW4T9URpYGJ3ym4+8zA05u44QjST8ZIoVtu3qE7fWmdn5LPdqv\ngcZz8Ww8BWJ8X3w0PhQ/wnCDGd+LvlHs8dRy6bLLDuKMaZ20tZrqisPJ5ONiCq8yKhYM5cCg\nKOu66Lsc0aYOtZdo5QCwezI4wm9J/v0X23mlZXOfBjj8Jzv3WrY5D+CsA9D7aMs2gGfjve8A\nrD6mePZSeCfEYt8CONWDw8FXTxrPqx/r9Vt4biXeANh8vV7/+/16ffMD1N8AuKD/A/8leAvF\nY9bLAAAAOGVYSWZNTQAqAAAACAABh2kABAAAAAEAAAAaAAAAAAACoAIABAAAAAEAAANIoAMA\nBAAAAAEAAANIAAAAAN/ryxkAAEAASURBVHgB7N0JvFx1ef/x50wWlgAhQMIOIUUWFQRRcQHE\ngigqrRYjqJVSFQQUZQugVir9ayuQECiKkqJCqygiKrYoLrUY2VxAAWXf930PIZDM+X+fuXNm\nfnMyd+7ce2fmnuXze72enGXOnOV97r2ZZ37LMaMggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggEDmBaLMn2F5TvA1utQp\n5blcrhQBBBBAAAEEEECgQAIv6lquLsL1kCBl4y56cvT7bJxKOc5iQ13mdtb88b/BYruvHJfO\nVSKAAAIIIIAAAv0S8M+0uU+SJvdLh/2OSiCpOVpT7/Lsm9IzgaP+yWzS55q7e+FtZmdc+j6z\nN69i0c+T9dtb9G/zrfr5ZJkpAggggAACCCCAQNcCU7Xlswqf5r6QIGXrFnpyRILU03sy5VEL\naorMpk13Y/323hUepmK2ka8P1zGPAAIIIIAAAgggUD4BfS6kIFBogcdbr666ri8vt3SLuniT\n1u1YQgABBBBAAAEEECijQBkTpBm60bMVWys2VkxTUIorkEqQrJYgLTRbGlv8RHLZsRkJUoLB\nFAEEEEAAAQQQKLFAWRKkHXWPz1Y8ovAPxXcqblJ4v/znFLcrzlLMVFAKJRCnEqSoliD5JWqI\nhnuDSyVBCjCYRQABBBBAAAEEyipQhgTpBN3caxQfUSxVXKm4WHG+4hLF7xSrKw5W3Kj4gIJS\nGIFqKkEaqkGqX15j4LrIommHmnntIgUBBBBAAAEEEECgxAJFH6Rhru7tiQpPhD6r8ESpXfHx\nnndVLFB8W3GX4goFJfcCy5QgtfyYN2qQYovu8xuflFWHmtk9mSwzRQABBBBAAAEEECifQNFr\nkN6tW3qHwqfDJUd+19UFxRYr9lI8qzhAQSmEwBnP6Pa+FFxKI0EyqzZqkPx1/TLQzC6AYhYB\nBBBAAAEEECijQNETpO11U71J3bIub67XHlyn8MEbKIURiJ4ILqWRIFVb+yDZJLNNg+2YRQAB\nBBBAAAEEECihQNETpAd1T3dSJA9iHekWex8UT6p8AAdKcQTCfkiNBEkJUUsNkuqQqEEqzj3n\nShBAAAEEEEAAgTEJFD1BOlcq2yguVOzcQSjpg+R9lXzAhh912JaX8icQJEjR2jr95Oc+lSDR\nxC5/t5YzRgABBBBAAAEEeivQ0nu9t7vOxN7O01nMUnxBsY/ifoV/KPYPzOqbYmsp1lFsrthQ\nsVxxtOJyBaUwAj7Ud2M4BiVHH1dN4VceVxO7+5JMaehSeVhsYW45F4IAAggggAACCIxRoPXz\n4Rh3kuG3+eALeiaobaf4rsI/JXtN0jsU+9en3qRuiWKBYo7idAWlWAKeEAdllVozu/m1+x43\nRq3TDwt9kAIlZhFAAAEEEEAAgTIKFL0GKbmnPpLd++sLXms0XaFRnWsPjn26vp5JcQVSCVKl\n0Q9Jl+w1it73zLNn+iA5BAUBBBBAAAEEECixQNFrkNrdWm9ad6/iVgXJUTuh4q1LJUhROkGq\nX3G0hp4W7MkzBQEEEEAAAQQQQKCkAmVMkEp6q8t82d4HKSxhghR5DVKjTKMWqWHBDAIIIIAA\nAgggUEYBEqQy3vXyXfOwCVLVql6b2Cj6hdisscAMAggggAACCCCAQOkESJBKd8vLeMHpGiQL\nm9jdGYroF2JOuMw8AggggAACCCCAQLkEij5Iw0G6nT4ow2jLFXrDlaN9E9tnVaCqGiQ9FrZZ\nGgmShvq+I/yWILKKEiStpSCAAAIIIIAAAgiUUqDoCdJhuqs7jOHOfl7vIUEaA1w237JcCdKU\n8NQaCZLSJh/hsFFii6lBamgwgwACCCCAAAIIlE+g6AnS3rqlP1C8QXGR4huKbsrN3WzENnkR\nuFYJ0uvDk20kSKeYPTTP4qWRRavVNyBBCqWYRwABBBBAAAEESiZQ9ATpId3Ptyh+rfBk6UTF\nHxX9Lt5qaw9FS7VFh4Nu2+E1Xhq3wKXLlSD58O715pbhKHa1nXs/pJfXD7NFfcoEAQQQQACB\nTgL+f/xfK1racHd6A68hMECBq3SsJwZ4vEIdqugJkt+sZYqPKK5RnKHYRdHvMlsH+J5iapcH\nSv64JtMu38Zm3QvEj+pRsPUEKZ6Vep83s6slSKpJWvNwi2fqB0XbUxBAAAEEEBhWwJOjSxTP\nD7sFLyAwMQKr6rBfUPzzxBw+/0ctQ4Lkd+kvis8o/kGxneJ6RT+Lf+CeMYoD+GASixRe80Tp\ni0DktYl/Vd/1TE3duurLSor8fjXKKkMj2ZEgNUSYQQABBBBoI+BfanpyNK3Na6xCYCIFfqGD\n86X7OO5AmT6QL5DT9op+J0fjuB28tX8C8cPNfUf6o3HMeslybNWWBCliqO+EhikCCCCAAAII\nIFA6gTIlSKW7uVxwi0CQIPn6Sesnr8apkey0nn5ICQ5TBBBAAAEEEECgZAIkSCW74SW+XG9i\nF5R4g2RhRSpBGnoWUvIqUwQQQAABBBBAAIEyCZAgtd7tQ7V4reKQ1tUsFUAgVYMUNWqQlpj5\nKHaNwrOQGhTMIIAAAggggAACpRMgQWq95f6h2fspNT48t77MUn4Fwj5IfhXNBEmjYzyvpCis\nYZqT3+vkzBFAAAEEEEAAAQTGI0CC1Kr3VS2+SvG11tUsFUAgTIB0Oc0mdn5tGpghHKhhk4O7\nf4ZVAWi4BAQQQAABBBBAAIFEgAQpkRiaejOs6xSp5litG7GUR4Fq6p42a5D8auJgqG8N+z1J\nY7Zunser5JwRQAABBBBAAAEExidQlucghUr+fKLpCj3uxp5TPKVQNxRKwQUeSV1fuhllWINk\n+sXwZna3pd7DIgIIIIAAAggggEDBBcpSg7Sj7uPZCv+Q/ITCO+XfpLhP4UnS7YqzFP4AUUoh\nBeYrCY6fbV5a3JIgVXkWUpOGOQQQQAABBBBAoMQCZahBOkH398T6Pb5H0ysVniR5YuQ1Seso\nNlOo24ntq/ik4jwFpXACkTezW3PosqLGMN++rG8KWmqQqlZRDVJ1aFP+RQABBBBAAAEEECiN\nQNETpLm6k54cXaL4rOIaRbuiPvq2q2KB4tuKuxRXKCjFEvAEacuhS4rX09RrUGtZ0HIlSFOG\nXqj9G1msBImCAAIIIIAAAgggUDYB/4BY5PJuXZzXDPh0uOTIrz9WLFbspfBmWAcoKIUTCIfy\njiaZHeNJUq0sNHtAQ30vS5aVMZMgJRhMEUAAAQQQQACBEgkUPUHaXvfSm9Q1PviOcG+f1Os+\nit3GI2zHy7kUqDWxC888bGbnSbL3TUsKCVIiwRQBBBBAAAEEECiRQNETpAd1L3dShK2nOt1e\nH+HOkyofwIFSOIH0w2IrLQM1qNYoSJCi6UcO9U8rnAIXhAACCCCAAAIIIDC8QNETpHN16dso\nLlTsPDyDPye01gfJ+yqtrvhRh215Kb8CXT8LyS9RWTW1SPm915w5AggggAACCCAwJoGiD9Lg\no9HNUnxBsY/ifsV9iscVzyjWUvgodpsrNlSor74drbhcQSmewEOpSwqb2KkjWvWOqDZuw9BW\nGr3BE6Q/pN7DIgIIIIAAAgh0L7CFNt2hi81f1DYXD7Pdalr/DsXLFGsovMXHZYqbFemyrVb4\nl+NevP/53bW59v/4/l5Zf+nnmuqRIKMq/gW793P3slThX7R3Kv5Z1D9736vg80UnqQl+zW9S\nkUusi1P/e7tI8UXFbop0TdLzWveAYoHidIX/0FIKKeBN7PxvWaO0NLFTdaoP6NEo2pIapIYG\nMwgggAACCIxJYG+96ytdvNP7gfuX1mHx/7SPUxyvmB6+oHl9j2nfVXxM4Y9uScoHNPNP9QV/\nBuZByQttpidrXZLgbKX5W9ts02mVPjrYD+ob+GdO/8J9uM+Rr9JrP65v61/gf7A+zySDAkVP\nkBJy/+D7/vqC1xr5L9mqCn9w7NMKSikEqkqQ/G9ZUqKWBEl/ae8IX9W2W/AspMSKKQIIIIAA\nAuMS+L7e/T8d9uA1SOniCdD7FF5j9HXFTxX3KTzp2l/hydBGircr0gNyecLyHsWhCm8hlC7+\nedD304vix/Jk7r2KhcPs0M+XkhOBsiRI4e3wpnUelNIJPKMEKfxyKt4gJFBV4h1eb98sPAup\nacEcAggggAAC4xLw5m7njmIPh2hbT468lc/OikcVSblJM19W/EzxFsWBirMUYblcC7so9lD4\nduniydMqihsUL0+/OMrlG7X9hgo/3+ESpP30Wi+Opd1Q+i3Q+oV5v4/G/hGYUIFFz+uRV/6c\nq3pprUE6s1ZFHzf+AOuroDnJlkwRQAABBBBAYGACXhvzaYU3o/t7ReP/Zs0n5SXNHKt4UrFt\nsjKYXqB5r9mZG6wLZ71G548KT7bGW7z264cKT+Q2bbMzX7+F4jttXmNVBgXKWIOUwdvAKQ1O\noPYspDXrx1s/fVz9JdVADTazvn6z3dWZ8tL2VfPpt7KMAAIIIIDAqAU0MtTeFavspv9/Jo36\nzf17w5LIqt86xez2/h2i455fr1c3U1yr+L8OW/pAB+sqPBFKl7u14iqF1xR5bVTYzG49Le+p\n+IzCj9WLcr528mGFJ2SnpnboyZgncpco/l/qNRYzKECClMGbwin1UyB+SM2Etxw6Qux/IL0W\n1b+hqpfoDs34Nz0q0eTXWLzppS3PR6q9wD8IIIAAAgiMW8CTo0lW+YnvyKtMslRiiw472OIt\nFpmp9UVPiteg7NZhT7foNf0fXStb16c31KedJu2So2R7T1pOU6Sb2e2rdZ6QflfRqwTpV9rX\nY4p0guSfM96n+IFimYKSAwG/aRQEyiSgfkhJifTH8VNJbVFtpf6D8gSpUfQLMqexwAwCCCCA\nAAI9FND/MTv1cHc93VVk0Sz1y92khzv10eR+3SF8COykbFCfuTlZMcZp0szOE5SweI2O91G6\nN1w5znmvobpQ4V+ybhbsa1fNb6SgeV2AkvVZapCyfoc4vx4L1JrYBfuc4s3sGknTMM9C+t/g\nDcwigAACCCDQEwF9ov7uZIsP92SkJzvs6U7in6qd2GiHve50BhfrxV902ODK4LV76vMtX2IG\nr3c76wM8/EbhQ3l7Mzvvt+SDKXhN1uGKXhevsfqYwkezS5rZeTLmNWOXKsY7GIR2QRmEAAnS\nIJQ5RoYEqvojpe/sGqXS0g9J9fTeB6lR9OBY1SBVG8vMIIAAAggg0CuBhWa3HWM+Ymq8tf6n\n8SZfmSgrNGiR2qXdpJPp1HxttOfqNTand/mmq+vb9SKh8KTFE6I9FJcovDbJr8trl3pdvIbM\nv3T1Y3iC5J+zPVny5x6JlZIXARKkvNwpzrNHApVGbVF9hxuEO9ZXS3esEqxQsqT/uCgIIIAA\nAgj0R2C+2RLt+Zr+7D23e/Waq+cUrxjhCvxz7I0KT+bmKl5QpIs3e/t3hb/uCdL+il8qHlX0\nuvg3qp54fUKxmWJbhfd3pnmdEPJUwq/S83TenCsCYxSIUwlS61Df+mrrvthir4KvFc2TICUY\nTBFAAAEEEBiMgCcaVyi86eG8Dof0pnNbKvy7zXbJkb/V/9/3mh3f1v9Pf73CB2foV/lefcc+\nEITXJN2luEpByZEACVKObhan2guBFd4OOCwtTez0gv9RvivYgAQpwGAWAQQQQACBAQn8k47j\nTeG+pNi7zTG31zoNslfbRhVxHcv5etWfFP9lhY8k588s6le5TDu+X7GfwpOyfiZj2j2lHwI0\nseuHKvvMsMDkVA2SbZA+WTWru0PrXubr1XF2HQ1zOl1/gZ9Ob8cyAggggAACCHQt8CFtufMI\nW5+g16+rb/N7TT+l8H5LFyuuVfxM4f+P++h/noD459hjFT9XdCrezO4rCk+0PDnq5//pSf+m\nI3QcLzSvG3LI1b8kSLm6XZzs+AWW6Q+r18Q3SroGSV9FRXcqSWoUDXPqtUh/bKxgBgEEEEAA\nAQRGK7Ct3uDRqZyRetGXfajv4xVvUeygSMqfNPNZxU+SFR2mj+s1H5H2bYpBJCzn6zieIN2g\nSBI+zVLyIkCClJc7xXn2SGDhUrPjnlXd0JpDO2ztg+Tr9PRw1SA1W59qWCESpB7psxsEEEAA\ngdIJnKkr9hhr8dohD31faZsqVlPcpnhG0a58Tis90uXt6RX15X2HWd/N6hXaKPxONXnPVcOs\nv36Y9cn7mGZEgAQpIzeC0xiowEM6Wj1BildqYqe68dRQ34xkN9C7w8EQQAABBBBYWcBHtbtx\n5dWsQaD3AiRIvTdlj9kXeFinWOtjpC9+fPhNry7ywRlqRV8H3RH+YsQ8CymhYYoAAggggECR\nBXbRxU3v8gKXaLtLu9yWzXImEH4OzNmpc7oIjFnAE6SkKDn61Ez1AW2s01dUd6ydvKppZPEW\nwSKzCCCAAAIIIFBMAR8Nb8cuL+0Wbbddl9uyWc4Emh0tcnbinC4C4xDwJnZBWbWlmd0ijW6j\n5x89kWygJnfeB4mCAAIIIIAAAsUWeL0uz0dy6iZIjgr8s0CCVOCby6UNJ5B+WGx1pZHs9E4f\n6jspszXD70qiwRQBBBBAAAEEECiwAB/6CnxzubThBCqN5nRDW7QbyS5qJEh6FtIUPYhhk+H2\nxnoEEEAAAQQQQACB4giQIBXnXnIlXQtUU03sKi1N7Hw3PpJduLspNLMLOZhHAAEEEEAAAQQK\nK0CCVNhby4UNL1BN1SCt3MQurj0LqbkHPeRgTnOJOQQQQAABBBBAAIGiCjCKXVHvLNfVQWBy\nKkFauYmdvjloqUGKGOq7gycvIYAAAgggMKyAjwS7w7CvNl94UbMXNxdb5lbT0jsU/ogOf2Ds\nnYrLFDcr0mVbrdimvvIaTe9ObxAs+/5eWV/2h9H60N2jKfr+1N5df4MeRG+XjPDmffS6f/a+\nV/EHhTfff63iQcVVik5lQ73og0j49fh1UfooQILUR1x2nVWBZUqQfICaRllpkIaqEqSwelVN\n7rZsbM0MAggggAACCHQrsLc2/EoXGz+pbdZJbecJyHGK4xXp5xPpv2r7ruJjCn+IbFI+oJl/\nqi+crelByQttpidrXZLgbKX5W9ts02mVf1T4QX0DfVSwzRWe/LQrr9LKH9dfOE/TDyr8PRco\nHldsrFiuGK4coxeOUnxIQYI0nFKP1pMg9QiS3eRJYKG+5Tn+GZ3xWvWzXqkPkr6euWe2xcs0\nQEMtk9KzkJJvo/J0oZwrAggggAACWRH4vk7kfzqcjNcgpYsnQO9T3Kn4uuKnivsUnnTtr/Bk\naCPF2xXLFGHx5OM9ikMV7RIP/wzg++lF8WN5MvdexcJhdujnmy73a8XPFF479laFX1+74p/X\nPaF6SnFhuw1Y11sBEqTeerK33Aj4UN9RkiCtnz5tfZ2z4lizW7S+9pwD/eXzb5b8j5//EaQg\ngAACCCCAwOgEvNbj3FG85RBt68nRA4qdFY8qknKTZr6s8OTiLYoDFWcpwnK5FnZR7KHw7dLF\nkyf/EvQGxcvTL45y+UZtv6HCz3e4BGk/vdbuWF7L5QnS3yt+qmhXPAFcX+E1cUvbbcC63gqE\nrYh6u2f2hkCmBaKHmqcXrWc2d1JzeWgutsj/ANeKapJWO2Ko6jxZxRQBBBBAAAEE+iPgX0h+\nWuHN6DxxCJMjLdbKS/r3WMWTim1ra1r/8aZr/qXm3NbVjSWv0fmjovF/feOV0c947dcPFTsr\nNm3zdl+/heI7bV7zWrVHFN7Uz/tXtSv/UF/pyRRlAALUIA0AmUNkUaBWg5ScmL4omD1TC0HS\nVKsuavmjqQzK/wDflbyJKQIIIIAAAuMXOM6bee2mz/IrfVE3/n2PeQ9LdD7fMjvl9jHvYXxv\nfL3evpniWsX/ddiVD3SwrqJd6467tf4qhdcUHaJYrkiKvhi1PRWfUfixelHO104+rPCE7NTU\nDvfXsidylyj+X+o1T/T+U3GMwpMkubeUGVrywR2uVvyp5RUW+iZAgtQ3WnaccYGHW8+v4lXX\nLQlS1ao3VaxZyaq5bbTNT1vfxxICCCCAAAJjFfDkKPrJ0Lu90iRLJT7M7GDVeix6vkdn5TUo\nSgSHLd6sPfl/eOv6Vt4kbaTSLjlK3uNJy2mKPRRhM7t9tewJ6XcVvUqQfqV9PaZIJ0j+QeJ9\nih8olinalW9opSdIXluWTpA8ufKmgP+hoAxIoPnpb0AH5DAIZEPAa5DCsvKzkPRf1U3hFmYV\nT5AoCCCAAAII9Epgp17tqPf7iWapxdcmPdyvjyb36w7htSRJ2aA+c3OyYozTpJmdJyhh8aTD\n+yjdG64c57zXUF2o8OZ0XvuVlF01s5GiXfO6ZJsbNXOFwmu1/AvbsHjzuucVnd4fbs98DwSo\nQeoBIrvIo0DYB8nPv5L8MW5cjNoX3BI2BmYkuwYNMwgggAACPRF4UTUYUw9XLZKSkayVWC0m\nTr21h2d1sfb1iw77uzJ47Z76/Mxg3VhmfYCH3yi86dohCm/OtqHCa7Lk3vPiNVYfU7xXkTSz\n82TsIcWlik6DQfgofW9U+PanK7z4F7OecJ2j8NF3KQMSIEEaEDSHyZpASx8knVyU/sbGztRz\nFeZZfK8GaKh1uFQdvv+hoiCAAAIIINAjgYW3qWXVHP0ftLXGI/AmXxkpk54zO/kmnUyn5muj\nPVevsUk++I/03qvrG3RKKEbaR/K6Jy2eEO2huEThtUl+XV671OviNWTeQsWP4QmSf872ZOk8\nxQpFp/I9veg+H6xPfVuvPfJy9tCEfwclQII0KGmOkzGBqv6AhS1MV25i5ydcb2ZXS5CUKM06\nwuK11Zj5qYxdDKeDAAIIIJBbgflqsMCDP1O3z2uulKTZK1Lr04v+OfZGhSdzcxUvKNLFm739\nu8Jf9wRpf8UvFY8qel2q2qEnXp9QeDO7bRXrKbppHufX68ncRxRbKZQ81/ok+fV5ckkZoED4\nCXGAh+VQCEy0QOzV3UFZuYmdvxgO9e3L+kvsf+woCCCAAAIIINA/AU80rlB408N5HQ7jTee2\nVKyiaJcc+Vv1hWit75Nvq9q62qAM39W0X8Vrgrz4QBBek3SX4ipFNyWpKfIkzmu9NlEk67p5\nP9v0SIAEqUeQ7CZvAo/7H8ywrNTEzl+MNJJduJHmaWaXAmERAQQQQACBPgj8k/bpTeG+pNi7\nzf6317pFCt9mfpvXw1Xna2EdxZcVPpKcP7OoX+Uy7fh+xX4KT8pGk4x5InWD4u8UXuP1ouK/\nFJQBC3jVJAWBEgqco2+ajvcOj2sNXXzcNkHSV1ga6rtZotpIdv7FFgUBBBBAAAEERiHwIW27\n8wjbn6DXr6tv83tNP6XwfjkXK65V/EzhX3D66H+egPjn2GMVP1d0Kt7M7isKT7Q8OXpa0a+S\n9G86on6AbprXhefydS0sUGyhuEjRj6aA2i2lkwAJUicdXiu6wEO6wHqCFG3Q7mKXq23z1JYX\n4m1aFllAAAEEEEAAgW4EttVGHp3KGakXfdmH+j5e8RbFDoqk/Ekzn1X8JFnRYfq4XvtfxdsU\no01YOux22JfO1yueIHltUJLwDbtx6gWvMfqSwj+f0LwuhTOoRRKkQUlznCwKPKyT8o6QXtZV\nbfYk9a1cMbQ49O9pZg9qJLtnNUDDmr5GgzaQIIVAzCOAAAIIINBZ4Ey97DHW4rVDHv7kjU0V\nqyluU3grkHblc1rpkS5vT6+oL+87zPpuVvtnBn00WKlcNcz664dZH+7Aa4xav5sNX2V+IAJh\n66GBHJCDIJAdgZaHxep3YfbMduemv3w3BuvnHGw2JVhmFgEEEEAAAQT6L+CjvPn/x9cohkuO\n+n8WHKEUAtQgleI2c5HtBVZ6WOz62u6h9LY+kp2SpNcNrY8mr27xlpoPk6b0W1hGAAEEEEAA\ngfwJ7KJTnt7laS/Rdpd2uS2b5UyABClnN4zT7aVA+mGx5v2QvBNoS4k1kp0GZ2is0y+NN7Mj\nQWqIMIMAAggggEAhBHw0vB27vJJbtN12XW7LZjkTIEHK2Q3jdHsq4H2QwuI1SO3KTamVniBR\nEEAAAQQQQKBYAq8v1uVwNWMVaH4tPtY98D4E8iuQak4XtU2QNJJdS4I0NNR3fi+aM0cAAQQQ\nQAABBBAYXoAEaXgbXim8QMsgDbra2JvYrVT0tLfbY4t9pJp6ibdN5pgigAACCCCAAAIIFEuA\nBKlY95OrGZXAilQTu/Y1SBfoSdYapOH2YNdbB/PMIoAAAggggAACCBRIgASpQDeTSxmtwOOp\nBMnaNrHzveqx2I1mdnom0lrHmm002qOxPQIIIIAAAggggED2BUiQsn+POMO+CZzzglKfp4Pd\nt21iN/R61EiQfLnKA2MDNmYRQAABBBBAAIHiCJAgFedeciVjEojCWqQONUjVlgRJvziMZDcm\nb96EAAIIIIAAAghkW4AEKdv3h7Prv0CYIK1rNnfSMIdsSZBiq5AgDQPFagQQQAABBBBAIM8C\nJEh5vnucey8EwqG+9fuw8ax2O9XjslsSpMhiEqR2UKxDAAEEEEAAAQRyLkCClPMbyOmPWyCs\nQdLOprRtZvdVsyc11PcjydE0aAMJUoLBFAEEEEAAAQQQKJAACVKBbiaXMhaBaliDpB1UNh5u\nLxrqO6xF2uQYs2nDbct6BBBAAAEEEEAAgXwKTM7naXPWCPRM4J7UnjZJLTcWY4tuVJK0m6/Q\nUN+ajf15SNc0NmAGAQQQQKBsAt5v9R1lu2iuN/MC62X+DDN+giRIGb9BnF7fBe5rPUK0aety\ncym26k2RNStd683sSJCaRMwhgAACZRLwJtrLFHqeOAWBzAl8LXNnlKMTIkHK0c3iVPshEKcS\nJBu2BinVxE6pko9kpyciURBAAAEEyihwtS56ehkvnGtGoOgCza/Di36lXB8CbQWevr91dTxs\ngvRSax8kY6CGVjmWEEAAAQQQQACBIgiQIBXhLnIN4xBY9Lze/Hiwg2ETpNPM7tFIdkuTbRnq\nO5FgigACCCCAAAIIFEeABKk495IrGbNA2MwuGnYUO+2+qmZ2tySHUQ3SVprndygBYYoAAggg\ngAACCBRAgA93BbiJXMK4BcJ+SGuYHT9juD1qJLvGUN8ayW6VI81mD7ct6xFAAAEEEEAAAQTy\nJ0CClL97xhn3XiBMkLT34fshpQdq0Piu2/b+dNgjAggggAACCCCAwEQJkCBNlDzHzZJAKkEa\nfiS7qob6Dk9cv0AayY6CAAIIIIAAAgggUBQBEqSi3EmuYxwCUdcJUroGKa4N9T2OQ/NWBBBA\nAAEEEEAAgUwJkCBl6nZwMhMjUL03ddxhR7J7WoM0aCQ7jc8wVBjJLpFgigACCCCAAAIIFEOA\nBKkY95GrGJ9AugZp0+F2t8jMhwW/J3mdZyElEkwRQAABBBBAAIFiCJAgFeM+chXjEngu9bDY\n4fsg+WHCZnYayW69Y8xmjevwvBkBBBBAAAEEEEAgMwIkSJm5FZzIxAmc+ZxGrnsqOP6wTex8\nm6pF1wXbmmqRXhUuM48AAggggAACCCCQXwESpPzeO868pwLhQA0dHxarJ8NWrw0PrV8iEqQQ\nhHkEEEAAAQQQQCDHAiRIOb55nHovBeJwoIa1zA5XtC9Vs5YESSPZbd9+S9YigAACCCCAAAII\n5E2ABClvd4zz7ZdAaqCGqcM2s3vG7GY1yXsxORGNZEcNUoLBFAEEEEAAAQQQyLkACVLObyCn\n3zOBVII0udNIdi+p39ENyZE1aMM2B5tNSZaZIoAAAggggAACCORXgAQpv/eOM++pQNgHyXcc\nD1uD5K9q9LqgmV00dQ2zl/t6CgIIIIAAAggggEC+BUiQ8n3/OPueCaxI1SB1Huo7Tg3UMNmM\nfkg9uxfsCAEEEEAAAQQQmDgBEqSJs+fImRKIwkEa/MxGqEGy1FDfFfohZep+cjIIIIAAAggg\ngMDYBEiQxubGuwonsDT1sNhKxwRpqbUmSGqSR4JUuJ8JLggBBBBAAAEEyihAglTGu841txE4\nQ4PTmUe9xMMO0uAbnGH2aGzxA8nWmtLELsBgFgEEEEAAAQQQyKsACVJe7xzn3Q+BsB9Sxxok\nP7hGr2sM1KBBG2bNM9ugHyfFPhFAAAEEEEAAAQQGJ0CCNDhrjpR9gSBBiqabHTOt0ylXLWrp\nh6QHyNLMrhMYryGAAAIIIIAAAjkQIEEym6r75B9sO34YzsG95BTHLRAHCZLvrHMzuyg1kp1+\nmUiQxn0P2AECCCCAAAIIIDCxAmVJkPYT85cVxym2rJPr0TV2vuIxxZ8U3v/kPxWqOaCUUyA9\nkt2Ujs3sVGPUaGI35MVIduX8ueGqEUAAAQQQQKBIApOLdDFtrsUTwB8q/iZ47XjN+zf9n1O8\nT/ErxW2KHRUfUmyh2E0RKyjlEkjVIFU7DtRwtdktr7V4mfofrTLEFDNQQ7l+XrhaBBBAAAEE\nECigQNFrkA7SPfPk6Jf16cc1fVLxv4qPKOYq9lB8TPE6hSdNuyj2V1BKJzC6h8VearZcRH9J\nmDRowzb6gfImmxQEEEAAAQQQQACBnAoUPUH6W92XJxT7KP5bcabiCIU3s/up4vuKsPyrFvyB\noZ4kUUonkO6D1PlZSM6j2qOgmV00WVVOrygdGxeMAAIIIIAAAggUSKDoCdLmulfehO6F4J55\n7ZG6j9gNwbpk1tffqdgsWcG0TALLU03sbJORrj5ODdQwiYEaRiLjdQQQQAABBBBAINMCRU+Q\n7pG+N6FbNbgLe2ver/vlwbpk1vtkvVpxV7KCaZkETntKXc+WBFfcRYK00kAN9EMKAJlFAAEE\nEEAAAQTyJlD0BOki3ZAZip8o3qP4tOI0hY9a54nSBxRJcYv/UPjodpcqKKUUCEeyi0dMkFaY\ntTwLSQmWDwBCQQABBBBAAAEEEMipQNETpEW6L54kvUXxA4X3MfJO9G9XnK34tuK3igsV3vfo\nQMUvFL5MKadA0MwuWsfsyNU6MSys9XFr6btEDVInMF5DAAEEEEAAAQQyLlD0BMn7FL1b8XeK\n+YrDFDsqHlYcqzhXMUfhr3tN0xkKH9iBUlqBlmRHCpHGXRixXJtsoUEb1jvSbONkmSkCCCCA\nAAIIIIBAvgS8z00Zyg91kR5hUX+TWo2RJ4k+mMM9CrWYopRcIKhBcomp3szulk4msUay0xDf\n70y20Q+U1yLdnywzRQABBBBAAAEEEMiPgCcHZSteUzRbsbXCv+n3JlQ+ch3JkRAoYR8k1xi5\nH5IGRWzph6RfqlfhiAACCCCAAAIIIJBPgbIkSDvq9nifo0cUTyg8IbpJ4bUFzyluV5ylmKmg\nlFpgdA+LdSq142w0sfPlyCokSA5BQQABBBBAAAEEcihQhiZ2J+i+nFi/N/doeqXCkyRPjKYr\n1BG/9tyjgzXdV/FJxXkKSikFRt8HST9Ut862eKn6H9UHdGAku1L+6HDRCCCAAAIIIFAIgaLX\nIM3VXfLk6BLFTgrva/RGxbsU+yv2Vuys2EjxZoXXLPnIdr4NpZQCo69BumCoeeafE67YbKsD\nW5+9lbzEFAEEEEAAAQQQQCDjAkVPkHwEuzsUPr2mw73QZ1pbrNhL8aziAAWllAILVbsYL21e\nejd9kLxZXdToh6T5SeuavaK5D+YQQAABBBBAAAEE8iJQ9ARpe90Ib1K3rMsb8qS28w+6DNPc\nJVhBNwtHshvxYbFuEFu1pR+SfrHoh1TQHw4uCwEEEEAAAQSKLVD0BOlB3T5vWjely9voI9x5\nUuUDOFDKK+APDa6XaD2NBr9qsjTclIEahpNhPQIIIIAAAgggkC+Bog/ScK5ux7cUFyq+qPit\nol2JtHIXhT9MdnXFjxTjKT7ww5cUU7vcycu63I7NBiMQ1iDpiDO8Fum2TofWGPHXTQo2iC32\nRJuCAAIIIIAAAgggkDOBoidIPhrdLMUXFPso7lf4h9/HFc8o1lJ4MrO5YkPFcsXRissVlPIK\npBKkVUZMkE4ze+pYi+9Rb6TNnE0ZN03syvvzw5UjgAACCCCAQI4Fip4g+eALCxUXKbwGaTeF\nj1oXlue18IBigeJ0RdC8SktjK0/obQeP4q0HaVtGzhsFWJ83TSVI3Q3UoB+2a5UY1RIkpUgz\njrJ401N78/PU58tl9wgggAACCCCAAAKJQNETpOQ679DM++sLXms0XeH9SvzBsU8rKAiEAqkE\nybwGqYsSXauNvKayVtTkzmuRepFw1/fIBAEEEEAAAQQQQKDfAkUfpKGdnzet8w+ttypIjtoJ\nlX5dNZ3UdJUgVVMj2Ylxh9JTAoAAAggggAACCORMoIwJUs5uEac7eIHqmGqQqis9ayt63eDP\nnSMigAACCCCAAAIIjEeABKlV71AtejOpQ1pXs1QugQWP6XpfCK5502B+2Fl1drtDo9f5e2tF\n/ZFIkBIMpggggAACCCCAQE4ESJBab9T6Wtxe4VNKqQViH/EwKV01sfONlRT9PnmTltb/VGPQ\nhuZa5hBAAAEEEEAAAQSyK0CC1HpvvqpF71j/tdbVLJVPIAqb2c00m9vlM62i34VWGgWFWqQQ\nhHkEEEAAAQQQQCDjAiRIrTfoYS1ep/AppdQCcTBQQ6SKoc037oYjtmpLghRZhQSpGzi2QQAB\nBBBAAAEEMiKgL7hLV2boiqcrVlE8p3hKsURBQSAUCGuQtD7yZnZ3hhu0m9dADb+bFLwQWUyC\nFHgwiwACCCCAAAIIZF2gLDVIO+pGnK3w5x75Q1z9g+5NCv8Q7EnS7YqzFGpKRUGgJpBKkCpd\nDdSgpw0/poEaGomUqp520t7K8nvGjw4CCCCAAAIIIJB7gTJ8cDtBd+kaxUcUSxVXKi5WnK+4\nROFNolZXHKy4UfEBBQWBVILU7cNiHS7shxStcYzZy+FEAAEEEEAAAQQQyIfA5Hyc5pjPcq7e\neaLCE6HPKjxRalf0Rb/tqlAFgH1bcZfiCgWltAItgzS4Qtcj2Xk/JPU92i+h0w+XN7P7c7LM\nFAEEEEAAAQQQQCC7AkWvQXq36O9Q+HS45MjvTqxYrNhL8aziAAWl1ALLg0EaHCLavFsO/TB5\nrWRQGKghwGAWAQQQQAABBBDItEDRE6Ttpe9N6pZ1eRee1HY+il1XI5Z1uU82y6XA/Ed02s83\nTz3eojnfeU4Z9jXqh7Qi2YqBGhIJpggggAACCCCAQPYFip4gPahb4J3kp3R5K3yEO0+qfAAH\nCgJe+1gvUdcJ0iIlVmpWFzap2+5Is9WSPTFFAAEEEEAAAQQQyK5A0ROkc0W/jeJCxc4dboM+\nz9b6IHlfJR+w4UcdtuWl8ggECZKtYXbMrO4vvWWghsn6AfORFCkIIIAAAggggAACGRco+iAN\n58nfP9R+QbGP4n6Fj072uOIZxVqKdRTev2RDxXLF0YrLFZTSC/hw3Z47J6UyR3Pe9G7EUh+o\n4aBkQ30T4QM1MPBHAsIUAQQQQAABBBDIqEDREyT1l7eFiosUX1TspkjXJHk/kwcUPoLd6YpU\n53ytoZRUIFYNUpgg1fohXdUNhg/UEL5To9opQap281a2QQABBBBAAAEEEJhAgaInSAmtN5V6\nf33Ba42mK1ZVeG3A0woKAm0EPEEKS+Q1SF2Vu83+MtviJZFF04beEHsNEgUBBBBAAAEEEEAg\n4wJF74PUjt+b1nkt0a0KkqN2QqyrC1THnCBdYOaj2F2TUCpR+quPm62bLDNFAAEEEEAAAQQQ\nyKZAGROkbN4JziqDAs/elTqprmuQht4XDtRQG/3jtan9sYgAAggggAACCCCQMQESpIzdEE4n\nSwKL1D8tfqh5Rt0P9e3v8YEamu+tzdHMLgXCIgIIIIAAAgggkDUBEqSs3RHOJ2sCQTO7eBOz\ng6d0e4IaErElQVIzOxKkbvHYDgEEEEAAAQQQmCABEqQJguewuREIEqRoktk0HxK+q3Ka2V2q\nR3o02JgEKcBgFgEEEEAAAQQQyKIACVIW7wrnlCGBSM9CCsvkOeFSF/NBLVI08wiz2V28h00Q\nQAABBBBAAAEEJkiABGmC4DlsbgSCGiQ/58oWoznz2OIgQTLTuPrUIo0GkG0RQAABBBBAAIEB\nC5AgDRicw+VNIP0sJBtHDZI/dtYfGEtBAAEEEEAAAQQQyKoACVJW7wznlRGB8SVIz5v9vvVC\neGBsqwdLCCCAAAIIIIBAtgRIkLJ1PzibzAmc/IBO6cXgtEZVg/QVs8fVzO725P2R2U5zzTTY\nAwUBBBBAAAEEEEAgiwIkSFm8K5xTlgSqOpm7mic0umchDb0vfGBstLqGwXtFc3/MIYAAAggg\ngAACCGRJgAQpS3eDc8mqQDhQwwyzI9YezYmmHxirWiT6IY0GkG0RQAABBBBAAIEBCpAgDRCb\nQ+VWIEyQdBFTRtXMTlVQLSPZMVBDbn8OOHEEEEAAAQQQKIEACVIJbjKXOF6BOPUspGhUCVJs\n9kc9MHZ5chbqk7RzMs8UAQQQQAABBBBAIFsCJEjZuh+cTSYF0iPZje5ZSAvNluqyrk0uTU3s\nXqkHxo6qmV7yXqYIIIAAAggggAAC/RUgQeqvL3svhEA6QRr1s5AstmhxkyKqTDHbpbnMHAII\nIIAAAggggEBWBEiQsnInOI8MC1RSTezGkiBVfxNeYGyV3cJl5hFAAAEEEEAAAQSyIUCClI37\nwFlkWuCkp9WH6IngFOcE813Nqh/Sb9T3SJOhUrF412SeKQIIIIAAAggggEB2BEiQsnMvOJNs\nC4Qj2W2mUx3V784Cs8f0nhuTS1SmtNPBZqsny0wRQAABBBBAAAEEsiEwqg952ThlzgKBCREI\nE6SpZsdtMvqzaPZDiiyaolEa3jD6ffAOBBBAAAEEEEAAgX4KkCD1U5d9F0kgTJB0XaMb6tsh\n9MBY+iEV6SeCa0EAAQQQQACBQgqQIBXytnJRvReIxj1QwzKzxa3nRT+kVg+WEEAAAQQQQACB\niRcgQZr4e8AZ5EMgVYMUbzHa0z7D7D4N1BAmWq9XPySN+E1BAAEEEEAAAQQQyIoACVJW7gTn\nkXWBVII0+qG+hy4wajSzUz+k1dYwe23WL5zzQwABBBBAAAEEyiRAglSmu821jkPgqnvUi2hF\nsIM5wXzXs+qHtDjceLIZw32HIMwjgAACCCCAAAITLECCNME3gMPnReDS5TrTe4OzHVOCFKX6\nIcUW7Rbsk1kEEEAAAQQQQACBCRYgQZrgG8DhcyUQNLOLZpkdM220Z3+K2a3qh/RQ8j4lTG/S\nPL+HCQhTBBBAAAEEEEBgggX4YDbBN4DD50kgDhKk2nmPeqCGoatt9kPScOHT55ltnycFzhUB\nBBBAAAEEECiyAAlSke8u19ZrgVSCVJkzlgOk+yHFZjSzGwsk70EAAQQQQAABBPogQILUB1R2\nWVSB8T8LyWWqZo2R7HxZo9mRIDkEBQEEEEAAAQQQyIAACVIGbgKnkBeBdBO7ypia2J1qdr36\nIT0VXDUj2QUYzCKAAAIIIIAAAhMpQII0kfocO2cCK1JN7Mb6LCSvRLLLkotXDdKsY822TpaZ\nIoAAAggggAACCEycAAnSxNlz5NwJLHhMz0J6NjjtOcH8KGfjlmZ2yphoZjdKQTZHAAEEEEAA\nAQT6IUCC1A9V9llggbAfUjymJnaOoyfOLm5FqtDMrhWEJQQQQAABBBBAYEIESJAmhJ2D5lcg\n7IcUrWY2b4OxXMtzZlerNur55L2RxdQgJRhMEUAAAQQQQACBCRQgQZpAfA6dS4E7Ws96bEN9\nLzJ7Sfu5MtmX+iFt/imzzZJlpggggAACCCCAAAITI0CCNDHuHDW/AqkEacwDNWikhvCBsWZT\n6IeU358KzhwBBBBAAAEECiNAglSYW8mFDEagemfqOHNSy10vVqy6ONw4MvohhR7MI4AAAggg\ngAACEyFAgjQR6hwzxwLVdA3SmAdqUBu7q/Q8JG9qVy/0Q0okmCKAAAIIIIAAAhMlQII0UfIc\nN6cCj9+lwRXi4OTHXIO00Gyp9vOHZF/qh7TN4WYzk2WmCCCAAAIIIIAAAoMXIEEavDlHzLXA\nOS+YRQ8ElzDmBMn3EVu0ONiXrWL2lnCZeQQQQAABBBBAAIHBCpAgDdaboxVCIA76IUUbmR2u\nvGZsJbLqr8J3VqyyV7jMPAIIIIAAAggggMBgBUiQBuvN0YohcGtwGfodWn2rYHlUs4/WHhgb\nq1ZqqKhPEglSgsEUAQQQQAABBBCYAAESpAlA55B5F4hubL2CaJvW5e6XzjHz5Og3yTvUD2nT\no83GvL9kP0wRQAABBBBAAAEExiZAgjQ2N95VaoH4ptbLr44roVGt0c/D/UVm1CKFIMwjgAAC\nCCCAAAIDFCBBGiA2hyqKQDWVII29BslFNCReS4JUsYgEqSg/KlwHAggggAACCOROgAQpd7eM\nE554gd9rkIbw+UXjaxI33+w61SI9lFyXEqbd55pNTZaZIoAAAggggAACCAxOgARpcNYcqTAC\nly7XpdzWvJxoq+b82OY03PcvkneqH9K0zc3emCwzRQABBBBAAAEEEBicAAnS4Kw5UrEEwmZ2\na5gdten4Lq+aambHcN/j8+TdCCCAAAIIIIDA2ARIkMbmxrtKLxCFCZI0Jo9roIaXzH6hZnZq\nXZcUhvtOJJgigAACCCCAAAKDFCBBGqQ2xyqQQHoku/EN1HC62cMave66AGhHDfe9XrDMLAII\nIIAAAggggMAABEiQBoDMIYoo0NsEyYWqFgXN7KKKEqY9iyjHNSGAAAIIIIAAAlkWIEHK8t3h\n3DIsEN3cenLxuJrYDe2LfkitpiwhgAACCCCAAAKDFyBBGrw5RyyEwElP6zIeDC5l3AnSY2aX\nqR/S0uY+47c255lDAAEEEEAAAQQQGIQACdIglDlGQQXCZnbRRmaHaTS7sZdzzF5Qs7rFzT1E\nmxxhtm1zmTkEEEAAAQQQQACBfguQIPVbmP0XWSA1kt0a465F0gNoG89DcrgpZnsVGZBrQwAB\nBBBAAAEEsiZAgpS1O8L55EggrEHy0x7fSHa+Bw33HQzUUNsnCZIzUBBAAAEEEEAAgQEJkCAN\nCJrDFFGg9wnSQrPrVYvU6NukJne7zzWbWkQ9rgkBBBBAAAEEEMiiAAlSFu8K55QTgUqqiV0v\nRrJTemRR0MwuWn0LszflBITTRAABBBBAAAEEci9AgpT7W8gFTJzASffq2M8Hx+9BHyQ11LPW\n4b5jq9DMLkBmFgEEEEAAAQQQ6KcACVI/ddl30QVi1feEz0Pa0mzupPFedNXsFxruW/tOSkyC\nlFAwRQABBBBAAAEE+ixAgtRnYHZfdIEoaGYXrWK2iVrEja/MN3tEe7g22MuOh5vNDJaZRQAB\nBBBAAAEEEOiTAAlSn2DZbVkE0gM1TO5JMzs1tGuMZhdZFK1qtmdZRLlOBBBAAAEEEEBgIgVI\nkCZSn2MXQKAaNrHT9VR6kiCl+yFpv28vABaXgAACCCCAAAIIZF6ABCnzt4gTzLZAeiQ760mC\ntNTsMvVDWpJce2TxOzXc97j7NyX7Y4oAAggggAACCCDQXoAEqb0LaxHoUuDFWzRQQzCgwvgf\nFusHPsNsmSaNZnZqcrfuZgz33eU9YTMEEEAAAQQQQGDsAiRIY7fjnQhIYKEqe+zuJkW8dXN+\nfHNViy8K9zDJKn8bLjOPAAIIIIAAAggg0HsBEqTem7LH0gm0jGS3ntnRip6Ui9XMbkVzT/Hf\nNOeZQwABBBBAAAEEEOiHAAlSP1TZZ8kE+jOS3QKzxwR5RRMz2vJos5c3l5lDAAEEEEAAAQQQ\n6LUACVKvRdlfGQWCZyH55femH1JtT6lmdpEZzezK+BPGNSOAAAIIIIDAwARIkAZGzYGKK1BN\nJUi964f0kllLPyQ9E4kEqbg/SFwZAggggAACCGRAgAQpAzeBU8i7wIupBKl3NUgLzW7TKHk3\nBkKvm2e2QbDMLAIIIIAAAggggEAPBUiQeojJrsoqcPrDSmKeCq6+J89CSvYXW9QAbbXrAABA\nAElEQVSoRVINUqQxxfdJXmOKAAIIIIAAAggg0FsBEqTeerK38goEtUjxFmZzp/aKYoVVGwmS\n71NJEqPZ9QqX/SCAAAIIIIAAAikBEqQUCIsIjFEgSJCiSWZzXjbG/az0tlPNfqsaKtVSDRUN\n1LDnwWarJ8tMEUAAAQQQQAABBHonQILUO0v2VGqB8FlIDtG7fkjaWaxmdv/d5I1WXcvsbc1l\n5hBAAAEEEEAAAQR6JVD2BEnf9NuWirV7Bcp+yiqQfhZStaf9kGylZnYVmtmV9UeN60YAAQQQ\nQACBvgqUIUGaJcGvKb4ZSE7X/FcVSxS3Kh5XXKfQczgpCIxFIJ0g9bQGyR41+6Uqkp5vnln8\nLs2X4fe3ecnMIYAAAggggAACAxAo+ges9WR4jeJjijl1zyma/kpxiMLnL1X8QOFJ03yFJ05F\nd9ElUnor8NQd2t/yYJ89rUE6x+wFjV7382T/GqhhPQ33/aZkmSkCCCCAAAIIIIBAbwSKngh8\nRkwbKz6t2KtO9glNX634D4W/9hbFXIV3qv93hSdOeygoCIxCYNFLquG5PXjD1sF8j2bjltHs\nlMfTzK5HsuwGAQQQQAABBBBIBIqeIL1BF3qn4mTFsvpF76qpP7Pm44qH6ut88qLiSMW9ij0V\nFARGKxCOZLemfpw8Ae9ZqZr9j5IwTZIS/20yxxQBBBBAAAEEEECgNwJFT5Ami+mPiuBDpa3Q\n8j0KfeO/UvHtHlD0bIjmlY7AigILpEeym9LTZnYLzB4T3hUJoJrZvUyd5np6jGTfTBFAAAEE\nEEAAgbIKFD1Bulo39q2KdYMbvFjzWylmBuuS2Q008xrFtckKpgiMQiCoQaq9a9tRvLerTWNr\nbWanYRipRepKjo0QQAABBBBAAIHuBIqeIJ0thlUUf1J40zovX1d44vQ9xUaKpOygGU+evKO9\nD9pAQWCUAi/9pfUN0fatyz1ZaumHpOcjkSD1hJWdIIAAAggggAACQwJFT5D+oMv0QRd8NLtf\nK7xm6MuKGxVvVtyluEHxsMKb4vkzkQ5XXK+gIDBKgWc9QQqbc75qlDsYcfNTNCy9apEaNVWR\n2c7Htib6I+6DDRBAAAEEEEAAAQSGFyh6guRX7s8/mq04STFDcYDiowp9tqwN8+3NoKYpvqvw\nb/z/Q0FBYAwCi/w5Rbc03xi9UvM9/x1T36MfBseoKCN7X3OZOQQQQAABBBBAAIHxCPT8w9t4\nTqaP7/UaIh/qezOFN7nbRLGzYjuFJ01rKN6v+LOCgsB4BLyWMimrmx3ltZI9LVWrejLfKEqY\n9m8sMIMAAggggAACCCAwLoGyJEgh0lpa8AfEPq14UtFuNDutpiAwFoH4utZ3Te15M7v5ZjpG\n7M1Ea0UJ0s5HDNWSJquYIoAAAggggAACCIxRoCwJ0o7y8QEbHlE8ofBnI3k/jvsUzyluV5yl\naDeynVZTEOhaIKxB8jf1PEHynaofUkst0lSz/Xw9BQEEEEAAAQQQQGB8AmVIkE4Q0TWKjyiW\nKq5UXKw4X3GJ4ncKNYWygxX+rfwHFBQExigQDyRBUgc6//ltFI1mRzO7hgYzCCCAAAIIIIDA\n2AWKniDNFc2JCk+EdlJsrnij4l0K/0C5t2JnhQ/37aPaec3StxW+DQWBMQicrFrJ2Gspk9KX\nGqSTzW7Wcf6YHETN7HbQaHZbJ8tMEUAAAQQQQAABBMYmUPQE6d1iuUPhU69FGq7EemGxYi/F\ns4oDFBQExioQ1iJtanb8jLHuqPP74pZapKpVqEXqDMarCCCAAAIIIIDAiAJFT5C2l4A3qVs2\nosTQBk9q4p3sN+5yezZDoJ1AmCD5632pRXox1cwuspgEqd3dYB0CCCCAAAIIIDAKgaInSA/K\nwpvW+ah13RT/pt+Tqpu62ZhtEGgvEKUSpLgvCdJptQcdx1cl56Bmdtsc1adkLDkGUwQQQAAB\nBBBAoOgCRU+QztUN3EZxocL7Gg1X/KGxuyouUfiADT9SjKfM0Zu9NmpJl3HGeA7GezMnkEqQ\nakl3X04yPZrdJJrZ9cWZnSKAAAIIIIBAeQQmF/xSz9P1zVJ8QbGP4n6FD+39uOIZxVqKdRQ+\neMOGiuWKoxWXK8ZT7tKb36fotubqbdr2kwpKIQSW3GA2zX+W6r9fUV9qkJxKD/H63lSLTzWL\n6l92xPtp9af9NQoCCCCAAAIIIIDA6AW85qQMxWt0vqjYTeEj1oXleS08oLhIcbriXsWgy0E6\n4CLFGgqvdaLkXuC4PytpeUX9Ml7Qo7Z0by9Y0Y/LmmfR/6l53e7JvqtWfb0eJvvbZJkpAggg\ngAACCCDQZ4Gp2r/3+feRoL3/f65L0WuQkpvjI9m9v77gtUbTFasqHlE8raAg0GsBb2aXJEj6\nWdtsay2rZqn3xZvZhQlSpdbMrkqC1Htq9ogAAggggAACJRAoeh+kdrfQm9Z5LdGtCpKjdkKs\n64WAJ0hBqfStmZ3GqFcfu9ib9NWKEiZv3lnG3+26ABMEEEAAAQQQQGDsAnyIGrsd70Sgk0Aq\nQZrUtwRpgdljSpJ+mZyMapM2mjc06EiyiikCCCCAAAIIIIBAlwIkSK1Qh2rRP9ge0rqaJQRG\nK1BNJUj9eRZSclZVNbNL5oemPDS21YMlBBBAAAEEEECgOwESpFan9bW4vcKnFATGIXDKQ3rz\no8EO+laD5Md4VkPTq2mdd46sFT00dt/dG6Po1VcyQQABBBBAAAEEEBhRgASpleirWvQPsl9r\nXc0SAmMRiP8UvGtDjSC/XrDc01kNgfi0dvjT5k6jmXpC8h7NZeYQQAABBBBAAAEEuhEgQWpV\neliL1yl8SkFgnAKx/ywFpX/9kIYO0trMTg+NPTA4OLMIIIAAAggggAACXQiUMUGaIZfZCh92\neWPFNAUFgX4IpPoh9e+BsX7yGsbuxxrNzmuS6iV+9xFmaydLTBFAAAEEEEAAAQRGFihLgrSj\nKM5W+HOPnlDcqbhJcZ/iOcXtirMUMxUUBHokMNiBGhaaLY0tCgZriFbVU9s+0KOLYTcIIIAA\nAggggEApBMqQIJ2gO3mN4iOKpQp/uu/FivMVlyh+p1hdcbDiRgUfKIVA6YXAM/7z9GKwp74O\n1ODHia36zeB4mo0+3LrMEgIIIIAAAggggEAngaInSHN18ScqPBFSn3XbXPFGxbsU+yv2Vuys\n2EjxZoXXLH1b4dtQEBinwKKXlLJ4klQv0bbKw6ckS/2Yzjf7rUazu6G572inI822ay4zhwAC\nCCCAAAIIINBJoOgJ0rt18XcofOq1SMOVWC8sVuyl0IjJdoCCgkAvBMJ+SGrxtpaSpP4WDfH9\njfAIU6xCLVIIwjwCCCCAAAIIINBBoOgJkj/TyJvUNZ4P08HCX3pS4SOP+eANFAR6IRAmSNpf\npe/N7Kpm/6WaK43ZMFRUo/T3aj/a15qr5FhMEUAAAQQQQACBvAsUPUF6UDfIm9Z1++Fwhrb1\npMoHcKAg0AOBOJUgRf7z1deiZnaPqErU+9nVSmTRemua7ZMsM0UAAQQQQAABBBAYXqDoCdK5\nuvRtFBcqvK/RcCXSC7sqvK+SD9jwIwUFgR4ILE09C6n2IOIe7LfzLtLN7CYxWENnMF5FAAEE\nEEAAAQTqAkVPkM7TdR6l2ENxlcKH9fapf7v+nfrUm+Ddr1iseLXiaMXlCgoCPRA441HtxGsy\nk9L3JnZ+IA3N+BM1s3s4OahqlN5+hNmGyTJTBBBAAAEEEEAAgfYCRU+QfPCFhQofxcufD+M1\nRV6T9A6Fj2LnU2/ytESxQDFHcbqCgkAvBYJmdtEss3kb9HLn7fZ1qZ4bq2ciqS/SUFEzu0ka\nIYLBRxIQpggggAACCCCAwDACRU+Qksu+QzPvV/jgC9MVmym2UqytmKZ4meIYxb0KCgK9FggS\nJN91/wdq8KO8ZNVv+LRZeCZS04I5BBBAAAEEEECgvUBZEqTw6p/RgidCtyqeDl9gHoH+CKxI\nJUiD6Yd0Wu3Bx7E3Ka2XaKt5Zm9KlpgigAACCCCAAAIIrCxQxgRpZQXWINBXgeqEJEhDlxR/\ns/XSeCZSqwdLCCCAAAIIIIBAqwAJUqsHSwj0QeCem7XTF5o7jnZozvd3bmmt7138fHIUjW73\nPrUl9WalFAQQQAABBBBAAIE2AiRIbVBYhUBvBS5Yof1dH+xzG7Nj1wyW+zZ7htkzVYt8mPt6\nidbQSCVzkyWmCCCAAAIIIIAAAq0CJEitHiwh0C+B3wY79t+71wTLfZ2tWDXVzC76aF8PyM4R\nQAABBBBAAIEcC5Ag5fjmcep5Eqjq0URhiTo9uDjccNzzJ5tdqmci+UiOtaIhv9+kh4MN5HlM\nyTGZIoAAAggggAACeREgQcrLneI8cy6wPKxB8mt53QAvKI4tPis83iSrfCJcZh4BBBBAAAEE\nEEBgSIAEiZ8EBAYicKqGlY+fCg41sBokP6ZGafi6jt8YKEKDNXzgULMZwfkwiwACCCCAAAII\nICABEiR+DBAYjEBsFgXN7KKNzI70BxcPpHzF7HEd/zvNg0Wra5SIDzeXmUMAAQQQQAABBBBw\nARIkfg4QGJxAqpndlIHWIsVW/XLrpUaHaZm/Aa0oLCGAAAIIIIBAyQX4cFTyHwAuf5ACK4Ia\nJD/u4AZq8KOdYnaN+iJd6fNDJZqjZyLtnSwxRQABBBBAAAEEEODbY34GEBigQJxKkAY6UEPt\nOpUgtdQiaUQ7BmsY4E8Ah0IAAQQQQACB7AtQg5T9e8QZFkZg/iMaKOGu5uVE/iykgf4O3m32\nfZ3Dw81zsLcdabZlsMwsAggggAACCCBQaoGBfjgrtTQXj0BNIAr7Ia1hdszLBwlzgdmLatq3\nKDmmapCiKVb5eLLMFAEEEEAAAQQQKLsACVLZfwK4/gELpJvZVQY6UINf7EtW1TOR4uXJhavZ\n3YHqizQtWWaKAAIIIIAAAgiUWYAEqcx3n2ufAIFqWIOk41cG+cDY2vUuNLs/tuiHycWrFmlt\nzf99sswUAQQQQAABBBAos8DkES5+M72+ygjbtHtZz1yxJ9q9wDoEyi2w/BqzSV57U//diwde\ng+T+VQ35Pckqc5N7UbFIzexi1SxREEAAAQQQQACBcguMVIP0Y/HcMoY4vNysXD0CwwksXKpE\n5Prg1VeaHbx6sDyQ2QVmi1vPI9puntmbB3JwDoIAAggggAACCGRYYKQapOTUn9fMrxX6cNdV\nubGrrdgIgVIK1AZq2HHo0qNJZtN30vxvBk/hQ35HQa2RD9ZQ9d9zCgIIIIAAAgggUFqBkRKk\ncyVzqOJlijcpfqT4juKXikYnb81TEECga4GqnodUOaS5eW2ghoEnSFWzb0cWn1Tvg2Saf4+q\nfjc5w+y+5rkxhwACCCCAAAIIlEtgpCZ26s9tWyn8eS0+NPDuip8qHlR8VbGbIlJQEECga4EV\nqYEaooEP1OCnOt9siX59v9k87WjyqlY5qrnMHAIIIIAAAgggUD6BkRKkRORqzaiLgs1W7KL4\nruI9Cm+Oc49Cn7XMmwlREEBgRIEFN6n/z7PNzSZmoAY/fmzVMzTM94rkXFSLdJCqjGcky0wR\nQAABBBBAAIGyCXSbICUusWYuV/ggDBsr9lRcovhHxR8UPqDDiYrNFRQEEGgvoNZt9vvmS5FG\ni/zU+s3lwc3pm407VYv0veYRozX0QKTDmsvMIYAAAggggAAC5RIYbYIU6vi3zv+rOEixgeKj\nipmKExSeMFEQQGBYgdpADcGrq+wcLAx4tnpyeED1SfrkgWarhuuYRwABBBBAAAEEyiIwngTJ\njfTNtx2t8A7m/6HwB07eq7hWQUEAgWEFfKCGsAz+gbHJ0U8x+5Ma2/0sWVaCNEvfdPAlRwLC\nFAEEEEAAAQRKJTCWBMmTIu/IfZXiboX3P/J1GvyqNtKdN6/7oYKCAALDCryYGqjBJrAGyR8c\nG7fUIqnZ3dFz9UTbYU+fFxBAAAEEEEAAgYIKdJsgbarrP1JxpeIuxQLFFgofyW53xSaKTymu\nUHg/JQoCCHQUOE0jQcbBcNrxa7V51PEtfXxR33L8SoM1eD/CWlEt0l/pm459k2WmCCCAAAII\nIIBAWQRGSpA+LAhPerym6FSFD/n9dcVbFRspvDP3rxVVBQUBBEYnENQiRdPVWnXr0b29t1ur\nFumkcI8Vi44Nl5lHAAEEEEAAAQTKIDB5hIv8pF5/leIxxYUKf0DsSwoNdGXvUgxXNIyx3Tzc\ni6xHAIGagPdDCmppJnkzO//dmZCiauEfHGvxbarI2nLoBKKdjrF4T9Uu+e89BQEEEEAAAQQQ\nKIXASAlSgrCeZj5Wj2Rdp+nn9eKJnTbgNQQQqKoGKezmUxuo4dwJdKmqFmm+ao6+lpzDUC1S\nTIKUgDBFAAEEEEAAgcILjJQg+ch0PoT3aMvi0b6B7REon8CSq83W1HD5UZIlTehADe6vquJz\nZ1msLzei9YfuR/TWoyzeUe1r/1i++8MVI4AAAggggEAZBUZKkL5SRhSuGYHBCJz5nNlxN+hY\n2w0dL97e7EA9f+icFwZz/JWPco7ZC2pWd7pqjv41eXWSVY5TN8P9k2WmCCCAAAIIIIBAkQVG\nGqQhvHbvl6CO5CsVb37nfZX8GUgUBBAYlUAcDtQwRc9a3mlUb+/Dxss1OqVGtHu2uev4vRrC\nck5zmTkEEEAAAQQQQKC4At0kSKvr8r+t8EEX9mhDsbvWna54QHGQgoIAAl0LVK5s3bTy5tbl\nwS+dZvaUmtgtSo6sIb8nTa7VIiVrmCKAAAIIIIAAAsUVGClB0jfa9nPFBxSPKJYp0sWbCJ1d\nX+kfqo5Ob8AyAggMJ7DCh8kPSjThCZKfzHKrLtRzml5snlj8j0eYzW4uM4cAAggggAACCBRT\nYKQE6R912W9SXKTwJjYXK9LFEySvOdpV8bDiXxQbKigIIDCiwCm3a5P7g83eqGcvj9Q3MNi8\nP7PKju6PLfpGsnfVIk2ZYpXPJctMEUAAAQQQQACBogqMlCAdoguPFYcqlo6AoBG5bJ7Cm+T9\nwwjb8jICCDQE4rAWaQ2z10x4PyQ/tRes+kX1RWrUGkcWH6Bf8L9qnDYzCCCAAAIIIIBAAQVG\nSpC20jV74vNgl9d+gbarKrbpcns2QwABi8IESR6TMtHM7gyz+1Rz5EP910ukmq3KCckSUwQQ\nQAABBBBAoIgCnRKkqbrgaYpbRnHhPjzxM4qZo3gPmyJQcoFqKkHKRj8kvykvWlXDfcfBsOPx\nB48y8y9OKAgggAACCCCAQCEFOiVI3kHbB2b4a0XU5dW/Utv5cN8PdLk9myGAgJ3sI0R6/716\nib3fX6ffzWTDvk81ot2D6ov01eRAPqKdnov0z8kyUwQQQAABBBBAoGgCI30IW6wL3kCxY5cX\nvnd9u+u73J7NEECgJhD2Q4r0vLHju/2d67ufapFOUi3S88mB1Bdpf41ot22yzBQBBBBAAAEE\nECiSwEgJ0ln1i/2mpiP1K/qQtvkXhQ/m8F8KCgIIdC0QJkj+pjgT/ZD8TPSQs4erFn3F54dK\nVJlqlc8nS0wRQAABBBBAAIEiCYyUIP1SF7tIsb3iDwofpc6b/8xQ+FDE/i3yexXekfs/Fasq\nDlc8qaAggEDXAumBGrLTD8kvIbbqyfr3ueRyNLrd3CPNtkuWmSKAAAIIIIAAAkURGClB8uv0\nhEddEWrDd+tDkl2meELhHbf9GUg+ct1HFY8r/lHxdQUFAQRGJXCSfpfix5pviXfRfLd9/5pv\n69PcAjOdW6SB7YaK+iJFUyz6fLLMFAEEEEAAAQQQKIpANwmSD9agL4vNn82i1jbm/Ys8QVqi\nWKzQMyXNm9f5yFbnKCgIIDB6gVhv8d+neonWMTvOa24zU16y6nzVHPkolbWiE36PqpR3SJaZ\nIoAAAggggAACRRDoJkFKrvOPmlHf7Fpzu3U1VUdy834SGvXXvqXwpImCAAJjFshuPyS/JH0T\n8oRqjvxLklrxWiTVKv1LsswUAQQQQAABBBAogsBoEqQiXC/XgECGBbLdD8nhNKLdqapFeipB\nVJK0j2qRdk2WmSKAAAIIIIAAAnkX6HWCtIZA9lTMyTsM54/A4AVO8uarTzaPG3nikZl+SH5e\n6oz4lBKkU5vn6CcYzc/aeYbnxzwCCCCAAAIIIDAagZESpN9oZ39qs8N3at0/tFm/pdb9QuF9\nkigIIDA6gao299+5pMxUP6SXJwsZmnotUvAw6Oh1x5jtl6Hz41QQQAABBBBAAIExC4yUIK2p\nPXtfo3TxfkeNvgjpF1lGAIGxCmS7H5JflaqLlihBOiG8wopF/6bhLlcJ1zGPAAIIIIAAAgjk\nUWCkBCmP18Q5I5Bjgez3Q3JcJUnf1LDk3iSwXqLZegiaciQKAggggAACCCCQbwESpHzfP86+\ncAK3e5PWxlDamt8to5dYXWGxxmcIS/RZPQ9gnXAN8wgggAACCCCAQN4ESJDydsc434ILXLBC\nF3hZ8yKjDcyO3bq5nJ05PTz2Z2pq9/PkjDRYw9qTrdLS9C55jSkCCCCAAAIIIJAXARKkvNwp\nzrNEAul+SBV/3lgmy3KLNT5D7INL1Et8mKqV/ipZYooAAggggAACCORNgAQpb3eM8y2BwIpf\nt15knNkESQ+PvT626JzkfFWLNEUDf5+ULDNFAAEEEEAAAQTyJkCClLc7xvmWQOAPV+sinwsu\nNKv9kGqnGFn1c6pFej45XyVJ+2qYyzcmy0wRQAABBBBAAIE8CUzu4mR9mO/PpLabrWUf0je9\nfsPUdiwigMCoBS5dbvb6K/S2vYbeGm1iNk/N1k65fdS7GsAbTjZ7YJ4eFqsn2jb6H022SF2U\n4jcM4PAcAgEEEEAAAQQQ6KlANwnSDB3xi8Mcdbj1w2zOagQQ6E7A+yFF9QTJ3zHJm9llMkHy\ns1ti1VOmWXSwao80qISX6PXzLH7/KWbfGVrmXwQQQAABBBBAIB8CIyVI/6bLWHcMl/L7MbyH\ntyCAQEOgqgRpUmNJtTFv0cI3ghWZmj1TTQKP0cNjlSAtap5YNP9Yi/9HNUzPNtcxhwACCCCA\nAAIIZFtgpATp/GyfPmeHQFEF7tKXDHOWqCZmWv0KvTZJrdgszuoV360EbrbFhylJ2sHPUdON\n9O/nzapHZ/WcOS8EEEAAAQQQQCAtwCANaRGWEciEwAUvKrn43+apRLPMjn91czl7cxeYrdDD\nYz+uZyMFSVz8SY0D/srsnS1nhAACCCCAAAIItBcgQWrvwloEMiAQX9J6EtW3ty5nb+lUsytU\nc3RO88yiyRWLvtJcZg4BBBBAAAEEEMi2AAlStu8PZ1dqgRWpBKmydx44llr1OLUEfLJ5rtFu\nqkX6++YycwgggAACCCCAQHYFSJCye284s9ILzL9TicbNAcPOZkesHSxncvYMs0fVzO6z4cmp\nVumUw83WCtcxjwACCCCAAAIIZFGABCmLd4VzQqApENYiaVCVqXs2X8ru3ClmZylJuiY5QyVI\nG6xqlX9JlpkigAACCCCAAAJZFSBByuqd4bwQqAlUwwRJa6JcNLPTiVaVIB2WGrDhE2pqtz03\nFgEEEEAAAQQQyLIACVKW7w7nhoAt1/OQ7IUmRPS25ny25+ab/VYJ3deTs1Qt0iSFHplUG648\nWc0UAQQQQAABBBDIlAAJUqZuByeDQFpg4VKtuTRYu7HZcdsFy5mefd6qx6sW6YnkJJUgvUkP\nRTogWWaKAAIIIIAAAghkTYAEKWt3hPNBYCWB9HDfUeaH+04uQeN7Px5Z/Olk2aeTLFqgpnZ6\nrhMFAQQQQAABBBDIngAJUvbuCWeEQErgpZ+2rojz0g+pdtonm52tWqTfN68hWrdiFQ12R0EA\nAQQQQAABBLInQIKUvXvCGSGQEjj1Fg33rSG/kxK9yeywNZKlHEx9wIaPKl4KzvV9x5q9O1hm\nFgEEEEAAAQQQyIQACVImbgMngcCIAuFodlPN1vjrEd+RoQ00YMN16n/0pdZTis481GxG6zqW\nEEAAAQQQQACBiRUgQZpYf46OQLcCqWZ2lVw1s/OLvNOqX1BN2F+aFxxtuIZVTm0uM4cAAggg\ngAACCEy8AAnSxN8DzgCBLgSe/T9t9GJzwzg3AzUk53yBzv8liz+sJKmarIvMDpxntleyzBQB\nBBBAAAEEEJhoARKkib4DHB+BrgTOfE6JxWXNTaPZZsdu3VzOx9xCs9/FFmkSlmjRYWozGK5h\nHgEEEEAAAQQQmCgBEiSzmcLfRoHFRP0UctwuBaKwH5K/J3fN7Pykl1v1c0r2bvN5L+qbtPk0\nq6T6Jw29xr8IIIAAAggggMCgBUgKzPRIFrtRsfag8TkeAqMTWJ7qh5Sf5yGF16nqo6X1Ue3i\n5vr4MDW127W5zBwCCCCAAAIIIDAxAkVPkLYX6xtGiI3r9K8Nttukvo4JAhkSmP9n1bzc1zyh\n6M1mR67WXM7P3Clmv1bd0VnJGasWKVKcfaRZLq8nuQ6mCCCAAAIIIJB/gcn5v4SOV/CfevVV\nHbdovhg2X/q8Vp/YfIk5BDIj8DOdyUfqZ7Oq2dTdNZ+qWaq/mvFJZNVj1R/pnUqMNh061Wir\nyRYtMKuqSxIFAQQQQAABBBCYGIGiJ0hfE6t3CNcHSfuxwpvSpctbtOJ1in9XLK2/eHl9ygSB\njAlEngwlCZJma83scpkgnWz27NEWf2ySRT9JkDWq3aFq83qxnpt0cbKOKQIIIIAAAgggMEiB\nMiRIvxHoeYq3Kn6p+LIi6PtgJ2nZEySvMXpCQUEgwwKxfoaj5TrB5Hf37Rk+2RFPTdVFP1Xf\no68pMTok2Vg1St84xuLtlCQ9kqxjigACCCCAAAIIDEqg6H2Q3PEvCk+AzlScrvAmSkm/I81S\nEMiTwElPK7+/KjjjrfI43Hdw/va0VY/WoA03J+uUIM1SfD1ZZooAAggggAACCAxSoAwJknsu\nU/hodXsqtlVcr9hfQUEgjwL/3XrS0b6ty/laWmT2fNXiDypJeik5cyVI7zo2qFVK1jNFAAEE\nEEAAAQT6LVCWBClx/JVmfGS7Xyi+o/CmdzMUFARyJFC9sPVko79rXc7fkpraXa0E6Z9bzzxa\noCQpdw/Dbb0GlhBAAAEEEEAgbwJlS5D8/jyp2E9xgOKdioMUFARyJHDK7TrZ64IT3sns6M2D\n5VzOqs/RSUqSvM9gvUSrq7/Vtw82m5KsYYoAAggggAACCPRboIwJUmL6X5rxIcC/r7hU0Wje\no3kKAhkXiFO1SFNyX4sk8Kqa2n1IfazUzyop0U7TrfL5ZIkpAggggAACCCDQb4EyJ0hue5di\nrsKH+n5WQUEgJwIrfpA60Vz3Q0quRU3t7laS9Ilk2aeRxccfbbZbuI55BBBAAAEEEECgXwJl\nTJC8z9Fshfdt8NHspikoCORMYP6fdcK3BCf9BrN5GwTLuZ1VU7tv6eS/27yAqFKx6DsaZWVW\ncx1zCCCAAAIIIIBAfwTKkiDtKL6zFf5cFX/W0Z2KmxT3KZ5TeJ+OsxQzFRQEciIQh7VI+l2u\nvCcnJz7iab5o1UPVH+neZEONareR4ttaLsvfrOTSmSKAAAIIIIDAgAXK8GHjBJleo/iIYqni\nSsXFivMVlyh+p1BncFNfcLtR8QEFBYEcCKT7IeV/NLsE/TSzp5Qg7ado9A1UgrTnsVbx32cK\nAggggAACCCDQN4GiJ0jev+hEhSdCGunLfKSvNyrepdhfsbdiZ8VGijcrvGbJv6X2bSgIZFzg\n5D9oQIN7gpPc3ezIdYLlXM+qqd2VSpCOb72I+HNqaufPM6MggAACCCCAAAJ9ESh6gvRuqd2h\n8KnXIg1XYr2wWLGXwgdrOEBBQSAPAmEzu8lmq/xNHk6623NUknSqkqQfNbev9Uc6T89H8i81\nKAgggAACCCCAQM8Fip4gbS8xb1K3rEs5f0aSP1/GB2+gIJADgWqYIPn5FmI0uxD+JYv/UTVl\n/kVHvUTqKxidv7uZEkIKAggggAACCCDQW4Got7vL3N5+rjPaVOGJUqMvQ4ez9BHu7lb4gA3z\nOmw30kueeO6h6PYBl2/Ttp9UrKFYoqAg0K2AftaOf0Abrz/0hlhfBsRKIE72mtDCFP0yvlpJ\n0RXqh7RKclGq9j35FKselywzRQABBBBAAIEJE5iqI3uFhHdT8cqJXJeifwN7ru7OtxT+UM0v\nKn6raFc8UdxFoRY9tQEbgiY97TYfcd1sbfE9hf+wdFMmdbMR2yDQRqCqhEg/r9HHhl6rJRDv\n1HwwTHabd+Vs1SlqIjvP4iN1nWc2Tz2ep/5Il+mX9r+b65hDAAEEEEAAAQTGJ1D0Jnbnieco\nhdfmXKW4rz69WNPvKHzqWe79isUKfUttRysuV4yneHMgr43yZyx1E4eP52C8t+wCxR3NLryz\nSpK+qloj/72tFdUmRYr/1C/4Vsk6pggggAACCCCAwHgFip4g+eALCxXbKfwbda8p8lHr3qHw\nUex86s3vligWKOYoTldQEMiRwO/+Tyfr/efqJdbP9YGrJktFmi6x6sEatOGm5JqUIK09yaKL\nDjabnqxjigACCCCAAAIIjEeg6AlSYuM1Ou9X+OAL/kFqM4V/67y2wmt4XqZQax27V0FBIGcC\nly7XCf+4edKRfqbXf1tzuThzal/33HKL36skqdHHSknSNtMt8trisvw9K84N5UoQQAABBBDI\noEAZP1A8o/vgidCtiqczeE84JQTGIJAezS7adww7ycVbVCX8FyVIH1J4DXGtKEl6xzFW+ddk\nmSkCCCCAAAIIIDBWgTImSJ2sDtWL1yoO6bQRryGQPYFHfMTG55rnFe9jdvCU5nKx5jQww0Ua\nnOKE8Kr0x+w4jXbnNcUUBBBAAAEEEEBgzAIkSK10PlSy90mqD5nc+iJLCGRX4JwXlDD8pHl+\nkZqPzvjr5nLx5jRowxd1zd9vvbLo6xplZafWdSwhgAACCCCAAALdC5AgtVp9VYuvUnytdTVL\nCORC4MLWs4w+2LpcuKW4avGBSpK81rdW1NRuNQ3a8KNP8SVHQsIUAQQQQAABBEYpQILUCvaw\nFq9T+JSCQM4EnvwfnXDYzO7vNPaID0JS2KKmdktWWPy3SpIebV5ktMkqFv1gbvfPIWu+lTkE\nEEAAAQQQKL1AGROkGbrrsxVbK3xUu0J/gPz/7d0JnBxltffxUz0JOwQCyA5J2Fdl3wTCVVRE\nwC0oiBJFA8gFgawuF0HhviQTNlEQBI1cFgHxIiDiBhFkv4DsO0mABAgECEsSkkzX+z+d7nR1\np3umZ6aX6qrf8/mcqbVr+db0TJ9+nnpK50dJjcAl83SqkVok780uoyQp2UX9889Qhw0jFIuK\nZxrsNcQyUzTtXftTEEAAAQQQQACBmgXSkiDtKJFLFbMVbymmKfxZKv7gWP/G/QXFxYq1FRQE\n2lng8tKDD75eOp3MqU6zf6oWSS3rikWZ0eFj6dmuCMIYAggggAACCNQkkIYEyXu6ekhxtGK+\n4h7FnxTXKG5V3K9YSaFnTdpTiiMUFATaVOCsqUoUPPHPl+ATZuPWL0wleagkye8h/Hn0HJUk\nTRhjdkx0HuMIIIAAAggggEB3AklPkHQbgp2u8ETIe7baRLGX4nOKryoOVOyu8A+Q+ym8ZulK\nha9DQaAdBbI66CsiB673eCbpnTUsPd1Jlv2emtrduHSGRtRxwy/Gmn02Oo9xBBBAAAEEEECg\nmkDSE6TP68RfVPjQa5GqlVAL7lB8SvGe4hsKCgJtKrD4f8oOPE2/z9m5FupZSKHXDOeKEqQO\nxTVKknYqzGOIAAIIIIAAAghUE0h6grSDTtyb1H1YDaBs/tuaflThnTdQEGhTgbOf1IE/GDn4\n7dTSzO/DS0W5xGyeuv8+WDVJXiOcL8EqSpJu1jOSNinMYYgAAggggAACCFQSSHqC9KpO2pvW\nDax08hXmeQ93nlR5Bw4UBNpYICyrRQpT0VlD4YKp++/ZgYUHKkl6qzBPje3Wy1hwy0lmeogu\nBQEEEEAAAQQQqCyQ9ATptzrtrRTe9bHfa1St6F5u20fh9yp5hw03KCgItLFA19U6+MXFEwiO\nMBvRUZxO/tgks2dUk3SokqSlNciqRdpmoAU3nmy2YvIFOEMEEEAAAQQQ6ItA0hOkq4RyikI9\nedm9Cu/dy4fei51/gPShN8GbqbhD4fcoqBWO3aWgINDGApNn6+A94S+UdcyG+T12qSp6RtK/\nlCB9Q+H3GeaKkqR9lCRdr24ra61ZLryUIQIIIIAAAgikQCDpCZJ/KDpXsb3idwqvKfKapM8q\nvBc7H3qTug8U+ixlwxTnKygIJECgvJldOjsfUXO7a9Vpw7jSCxocOMgy3gwx6X8DS0+bKQQQ\nQAABBBDoUSAtHw68Jzv1bJXrfGGQhhsrtlD4vQgrKzZX6HEp9rKCgkBCBF6/UYnB3MjJHGp2\nwmqR6dSM6hlJk/VtyVnRE9a3JV/Rg2Qvis5jHAEEEEAAAQQQSEuCFL3S72rCE6HnFNEPj9F1\nGEcgAQJTFihBuq54IoHuu1n5y8XpdI11Wvb7SpJ+GT1rJUmjlCRNjM5jHAEEEEAAAQTSLZDG\nBCndV5yzT5lAeHnZCafpmUhlp26mJOl4JUl+/+HSoiRpnJ6RNGHpDEYQQAABBBBAINUCJEip\nvvycfPIFOv+lWqRpxfMM91U/JJsUp1M3pgfJZo+SiXfQsrQElvl/uknp2KUzGEEAAQQQQACB\n1AqQIKX20nPiKRFQhUlwRfFcA1WYdBxZnE7fmB4ku2iRhSOUJN0RPfvQggtVk/St6DzGEUAA\nAQQQQCB9AiRI6bvmnHHqBLq8t7ZIyYzUhBKl9BZ1bTl/voUHq/vvhwoK6v5bJsGl6q3lm4V5\nDBFAAAEEEEAgfQIkSOm75pxx6gQ61SFJeHfktDczG3tAZDqVoxeYvasHyX5aSdKTBQBPkjIk\nSQUOhggggAACCKRSgAQplZedk06fQFjSe5se//Pd9Bkse8Z6+NmbCy38j2iSpFqkjCdJam43\nctlXMAcBBBBAAAEEki5AgpT0K8z5IZATmO3dfc+JYHxOnbdtGJlO7aieDP26J0mqZXuqiBBk\nVJt02WgzdehAQQABBBBAAIE0CZAgpelqc64pFvBnImV/XQQIOlRTckxxOt1jniR9aOH+5UlS\nhwW/Vk1SqrtGT/dvBmePAAIIIJBGARKkNF51zjmlAuHFSgDUq93S8m09J3Xg0qmUj3iSpKZ2\n3tzu6SJFribpN+q44ejiPMYQQAABBBBAIMkCJEhJvrqcGwIlAp0vqNboL8VZwbpma3yhOM1Y\np9lrSiL3r5Ak/UpJ0kkIIYAAAggggEDyBUiQkn+NOUMEIgLZiyITPnpc2XTqJz1J0nOSvCbp\nmQLGkt7tMuequd2phXkMEUAAAQQQQCCZAiRIybyunBUCVQSm/Uk1JC8VFwbD1VnD1sVpxlzg\nPLNX1XHDfkqSHo2KBJY5fZxl1PkdBQEEEEAAAQSSKkCClNQry3khUFHgui7N1r1I0UKX31GN\nwrjfk/S+hcOVUN5bmJcfnjLWMr/SOH8/y2CYRAABBBBAIAkC/INPwlXkHBDolcCCy/Shf1Hx\nJeHXzcasXJxmrCCg9ohvK0k6QDVJtxXm+TAw+7aSpKtGmdHJRRSGcQQQQAABBBIgQIKUgIvI\nKSDQO4HzX9f61xdfEwwyG/C14jRjUYELzd5fYOFnlSTdFJ2vJOkrq1vwR3XeQHIZhWEcAQQQ\nQACBNhcgQWrzC8jhI9BHgbLOGkI6a+gG8gKzDx+w8IvqI/2q0tWCA9WBw1QlSR8pnc8UAggg\ngAACCLSrAAlSu145jhuBfglMvEPN7J4obiL4mJrZ7VmcZqxcYKrZ4k7Lfl1JUsk9XEqQdslY\ncI96uNu8/DVMI4AAAggggED7CZAgtd8144gRqJdAWS3SAGqRepbNKkk6Vqv9tHTVYJjuTLpb\nNUm7l85nCgEEEEAAAQTaTYAEqd2uGMeLQN0E5v2PNvV+cXPhYWYnrF2cZqyawCTLnpq17Cjd\nl+S9AuaKapLWUk3SbUqSDi7MY4gAAggggAAC7SdAgtR+14wjRqBOAhe8q2Z2VxY3FixvttJ/\nFqcZ605gspm6+g4/r5hXXC9YSYnS/ypJOqY4jzEEEEAAAQQQaCcBEqR2ulocKwJ1F1j8M33A\n1201hRIcT5ffBYueh51mNy+ycH8ZvlFYWwlSR8Yyv8w/UJa/sQUYhggggAACCLSJAP+82+RC\ncZgINEbg7Ce13Zsj215Tzz89OjLNaA8C55rdryRpLzW3e6Fs1VPGWXDjOLNVy+YziQACCCCA\nAAIxFiBBivHF4dAQaJLApNL9ZE4xGz6gdB5T3QkoSXpez0pSL4Dh/aXrBQeF6rzhJLMhpfOZ\nQgABBBBAAIG4CpAgxfXKcFwINE1g4r/0wf7uyO42MdvtK5FpRmsQ0LOS3lBN0nC1V7wmurqa\n3G23nAX3qxvwvaPzGUcAAQQQQACBeAqQIMXzunBUCDRboLwWSS3DKL0VUE3SfHUDfrhed7qa\n3EXv7VpbidJtSpK+0dttsj4CCCCAAAIINFeABKm53uwNgZgKTLxRtUhPRQ5uB7Oxn4lMM1q7\nQKhuwE+T5+FKkuYXXxYsF1jmt2MtM3mEWUdxPmMIIIAAAgggECcBEqQ4XQ2OBYHWCai2I1Sn\nbNGSGR+dYrx3AsK8JmvhfnJ9NfrKwGz0EAv+doIZz5yKwjCOAAIIIIBATARIkGJyITgMBFov\nME3PRApnFY8jGG42brfiNGO9FTjb7AHdl7SrapIeir5Wze32X8GCB0eb7RqdzzgCCCCAAAII\ntF6ABKn114AjQCAmAtct1IHoNppoCbgXKcrRh3GBzpxr4T6qors6+nIlSRtlLLhTD5WlW/Uo\nDOMIIIAAAgi0WIAEqcUXgN0jEC+B8GLVIs0tHlPwBd2LtHlxmrG+CFxiNk+dNxyRtezJ8l1c\n2IaSpOX1UNlLdV/SxbovabnCfIYIIIAAAggg0DoBEqTW2bNnBGIoMOk9HdRFkQPT34gOVXJQ\n6iEw2ew8Nbf7pGJ2dHu6L2nUUNUm8bykqArjCCCAAAIItEaABKk17uwVgRgLZM9XLceHxQMM\nj1It0rrFacb6I6DOG/652MKdlCTdV7qdYLeBFjysNo2fL53PFAIIIIAAAgg0U4AEqZna7AuB\nthDofE2HeXnxUIPlzYJTitOM9VfA70uabuG+ui9Jre+KRU3uVjfL/K+a3J1Hk7uiC2MIIIAA\nAgg0U4AEqZna7AuBthFYpNZgli0ebuZ4apGKGvUYu85soe5LOia07DdVYzcvuk01ufueugL/\nl9o2Do3OZxwBBBBAAAEEGi9AgtR4Y/aAQBsKnPOsDvqayIGvpJqNH0SmGa2TgJrcTela0hX4\nk9FNqjZpV/Vy97C6Av9idD7jCCCAAAIIINBYARKkxvqydQTaWKDrx6rZ6IqcwCizUzaKTDNa\nJwE9L+lJdQWu5yXZlNJNBoM6LHO993InfCWpFAQQQAABBBBotAAJUqOF2T4CbSvQ+ZwOfUrx\n8P1epAGnFqcZq6dAvivwb6rJ3VHqwOGD6La9l7vVl3TgsEt0PuMIIIAAAgggUH8BEqT6m7JF\nBBIksPinOpmFxRPKjDQ7ebPiNGP1FlCTu8sXLWly93jptoMtQgvu1n1J39d8/naX4jCFAAII\nIIBA3QT4J1s3SjaEQBIFzp6hs1LlxtIywGz505ZOMdIQgfPMnlJX4Lupyd0vojvQfUkD9WDZ\n/x5nwe3fM9s4uoxxBBBAAAEEEKiPAAlSfRzZCgIJFljw37oXaX7kBA83G79tZJrRBgioK/D5\n6uXuP9Xk7iD5v166i2Df5S14VLVJR5bOZwoBBBBAAAEE+itAgtRfQV6PQOIFzntVH9B/HjlN\n/7vxk8g0ow0UUJO7W+ZbuL3uS7q5dDfBINUm/c9YC25QbdI6pcuYQgABBBBAAIG+CpAg9VWO\n1yGQKoEPJipJeq94yoG6np6wc3GasUYKXGD2RqeFB6s26btKlKK1eaZmd4eqNukJ1SZ9tZHH\nwLYRQAABBBBIiwAJUlquNOeJQL8EfjFHL1err2gJvQMHShMFVJt0UdbCnZQk/V/pboM1VZt0\ntWqTfn+C2dqly5hCAAEEEEAAgd4IkCD1Rot1EUi3wDmqRXqrSBAcaDZ27+I0Y80Q0DOTnn7A\nwj1Vm/Rfuh6RHgbNa5O+tKJqk8aajWjGsbAPBBBAAAEEkihAgpTEq8o5IdAQgYlztVlVYkRL\nx5nRKcabIzDVbLEuxBmqTdpVSdLDpXsN1g4sc63fm3Sy2Qaly5hCAAEEEEAAgZ4ESJB6EmI5\nAghEBLp0O4xFe1Tbz2yc7keitEJgstmj71i4u/Z9uprdLYoeg9+bNMCCJ1WbdJzm61mzFAQQ\nQAABBBCoRYAEqRYl1kEAgbzA5A/MsmeUcgT6nH7C8qXzmGqWgB5StWiSZU/rUqKkJOnR6H6V\nJK2m2qQLVZt050lmW0eXMY4AAggggAAClQVIkCq7MBcBBKoK3P9LNet6qrg4GGq24inFacZa\nIaAbxB6ea+Eufm+SEqUPo8egRGnv5Sz49xjL/Fg3Jy0XXcY4AggggAACCJQKkCCVejCFAAI9\nCkxdrFokVUhES+YHZietF53DePMFvDbJ700KLPyoktg7So8gWE5/8E8basFj6hL8k6XLmEIA\nAQQQQACBggAJUkGCIQII9EKg86/6AH5T5AWrmK3w/yLTjLZQYJLZM5MsHK5E9hjVJr1TeijB\nFuoS/G/jLHMNnTiUyjCFAAIIIICAC5Ag8XuAAAJ9FMiO1gsXFl8cfsNstHpVo8REIFSidMki\nC7dRknR9hWM6TJ04PKXapFOGmw2osJxZCCCAAAIIpFKABCmVl52TRqAeAp3PqRbpZ8UtBeop\nbeD5mqbHtCJKy8fOM3u108Iv696kg5UoTYsekO5NWlW1SWfvZsFD6u1un+gyxhFAAAEEEEir\nAAlSWq88541AXQTm/VRJ0uzIpvZUt99HRKYZjYmA7k26ebGF2+pwflreiYNy2u3V290dYy3z\nu++ZbRyTQ+YwEEAAAQQQaIkACVJL2NkpAkkRuOBdfbj+YenZZCaajVqpdB5TcRA412y+ugQ/\nVUnt9kqSdB9ZaVHV31fU293Tuj/ptFFmXMNSHqYQQAABBFIiQIKUkgvNaSLQOIGzfq0P3A9H\ntr+B2eAJkWlGYyag2qTn1Ozu012WVa/f4SvRw1OzuxU1/eNBSpR0f9JXo8sYRwABBBBAIA0C\nJEhpuMqcIwKNFciqtzS1zCopuqVlwpCSOUzETuBss99nLdwqVNfgSpQWRA9QidJGuj/paj1k\n9l9KlHaPLmMcAQQQQACBJAuQICX56nJuCDRNoPNOfcC+NrK7FTR+UWSa0ZgKTDb7oFMPl11o\n4da6hr8vP0wlSnsrUbo33y34sPLlTCOAAAIIIJA0ARKkpF1RzgeBlgksHqddz4vs/jPqsOFr\nkWlGYyyg3u6m69lJI7KW3V+J0iMVDvWwgeoWXB05nKPnJw2usJxZCCCAAAIIJEKABCkRl5GT\nQCAOAmfP0FGoA4BoCfS5e/Ra0TmMx1tANUpTp1m4sxKlY9WRQ7SHQh14sJw6cjhZz096QW0o\nx5xgtny8z4ajQwABBBBAoPcCJEi9N+MVCCBQVeAFJUT2YHFxoORogDpPo7STwHVmXUqULg4s\n3EzHfaYSpfnR41ezu9XVLXjnChY8p/uTjlZPDx3R5YwjgAACCCDQzgIkSO189Th2BGIncF2X\nPlt/W4e1uHhowZFm4z9dnGasXQQmmb2nbsF/pOcnba5j/o2a3qlDjmLJd+Rw6VALnlSidJiW\nqIKJggACCCCAQHsLkCC19/Xj6BGIoUDnv/VBWhUQJeWXapG1cskcJtpGQFWAM5UofUs93u1Y\n6flJyou2UEcO14yz4MHRZge2zYlxoAgggAACCFQQIEGqgMIsBBDor8Drp2sLzxe3EgxRKyx1\nJU1pZwFlvY/685N0f9IBSpQeWPZcgh07LHOLEqU71WPH/ssuZw4CCCCAAALxFyBBiv814ggR\naEOBKXqmTnZU6YEHJ6rDhl1L5zHVjgJKlP6uRGk3PWj2S0qUnlz2HIKPm2VuU6L0TzW9G77s\ncuYggAACCCAQXwESpPheG44MgTYXmHS7mtpdFjkJ/b0ZeKnZqIGReYy2sYAeNPsHJUrbK1Ea\nqWs9fdlTCfZV07vb9bBZhe237HLmIIAAAgggED8BEqT4XROOCIEECQT6XBy+FjmhHczWGB+Z\nZrT9BbJKlH47zcIt1fROPX+Hr5afkjpzGK5e76YqUVKtEk3vyn2YRgABBBCIlwAJUryuB0eD\nQMIEznpbH5j1obmk6FlJE3YumcNE2wuoa/CFanr389kWDgste5Ka3kUT49z5KVHSfUmZ25Qo\n3a2mdwe1/UlzAggggAACiRQgQUrkZeWkEIiTwKTfK0n6Y/GIAjWxC69UU7uVivMYS4rAFLMF\nnWbnq2twT5RO0bV+vfzclCjtqaZ3NytRelhVjCO0nP9F5UhMI4AAAgi0TIB/Si2jZ8cIpEmg\nyztsiHxQDrZUU7tz0iSQtnNV1+DzlSidu8jCoWp6p96/KyZKH1PTu2uVKD2hRGmkfkm4Py1t\nvyicLwIIIBBDARKkGF4UDgmB5AlMnq0HyH6r9LyCY8zGHVI6j6mkCXiipKZ356jp3RC/R0lN\n714uP0fVKG2lROk3gyyYpkRpjNpkrla+DtMIIIAAAgg0S4AEqVnS7AeB1At03qJahF+UMmTU\nq93YdUvnMZVEgSlqeuf3KM21cFM1vTtaidJz5eepRGkDJUqdK1rw0hjLnHWS2Xrl6zCNAAII\nIIBAowVIkBotzPYRQCAisFAVBOFTkRlr6wGyv4lMM5pwgUvMFqnp3a+nW7i1apQO1+/DY8ue\ncjBI/5zGL2fB9LGWuUwdOmy37DrMQQABBBBAoDECJEiNcWWrCCBQUeDc+XqA7BFatDCy+DP6\nLFze011kMaNJFLhObS5Vo/S7SRZ+VDVKB6lG6Z/LnmewXGD2LXXo8JjuU/qbsuvPah3NoiCA\nAAIIINA4Af7RNM62N1v+jlbWF6u2iuKD3ryQdRFoT4HxqhQIVJGwtCxQTcIuZhOfWDqHkdQJ\nnGy22wALxukf0xf0+1HxCzwlUk8HFp7/jtnl+qM5L3VInDACCCAQT4HldFgfKvZS3BPPQ6z9\nqEiQardq5JokSI3UZdtxFNDfnvF/14fg/ygeXPiIOrrbw2yKkiVKmgWUKG02wDLe891Rui9p\nxUoWSpTeCi34lWqfLjrbbEaldZiHAAIIINA0gUQlSBW/oWsaZfN31NP5duiQ1lCs0PxDY48I\npEog9A+/OmM9SLZQgo+arXNhYYphegXU893znZY9LmvhxkqA/ku/K6+WayhxGqw/6OMzFryg\n5nc3jDP7RPk6TCOAAAIIINAXgZ4Shr5sM26vWUcHdI3iLcW7itsVeysqle0109cbX2kh8xBA\noJ4Ck17R1tTVd7QE3zSb4M9MoiBgqhl6U+0wz5imLsKVKCmhDh8uZ1Gi1KE4VM+a/fs4C57U\nfUrHK1latXw9phFAAAEEEKhVIOkJkt/T84DiMIXXDvkHsv0UdyjOVFAQQKClAmfpXv3wgtJD\nCH9mNnrX0nlMpVlAvyQLlShdrg4ddlLPd/ured0f9XuTXdYk2FrdhP9cTe9mqve7i04xU60k\nBQEEEEAAgd4JJD1B0peJtpHidMWGiq0U/sHrccUPFOcoKAgg0FKBt/1ek7uLhxAsbzbg96oI\nWLM4jzEElgio57upnRZ+vsvCYaHZJP3uzCm3UY3SqrrJ7Vjdx/Rv1Srdo1+wo3RfU8V7mcpf\nyzQCCCCAAAJJT5C8J43ZijMU7+Uv94Ma7qu4U6H/meZJFAUBBFomcMkifcgdod2/XjyEYGN1\n6ni1ppP+N6p4yoz1SkDN72boPqXxsy3cUM3vu8G7jgAAQABJREFUjtbv0DLN75ZsMNijwzJT\nBqpWaZxlztUf/W17tSNWRgABBBBInUDSP3xsoCvqidDisis7V9OfUzyqmKjwJngUBBBomcCk\nWdr1V/Uht6t4CMEBuh/pJ8VpxhBYVmCK2QI1v/u1N79bbNm9Vat0pZrgeVezZSVYQzNOGmiZ\nx9Wpw936Zuxb313yaIWy9ZhEAAEEEEi7QNITJO/69ZOKSr3SeYcN/tBBvy/pt4pqHTdoEQUB\nBBovcNZU7WNC6X5CNYUdf3DpPKYQqCygNtN3q1bpyHkWbqBaJeVA4fOV1lQTvD11r9Jlq1jw\nqu5V+pUeyrV7pfWYhwACCCCQToGkJ0j/0GUdpPhvxfoVLvFMzdO31Lnmd7doeFCFdZiFAAJN\nE5ioW0zC64u7C3Qrif2PWsJuXpzHGALdC/zCbI5qlSarVmkLdepwgGqU9DsVlrck0EaCVfQL\n9u2MZe7VvUqqWbIxinW73zpLEUAAAQSSLuAfPpJcvOboQcU2Cu/x6GuK3ynKy8c043bF6vkF\np2t4Wn68GYPvaCeXKFZRfNCMHbIPBOIrMG5V3Xp0v47PO1UplGf1gO49zc71bvgpCPRawBMf\n/cM7Sj3cHa0apKoJt5KpLq13qzqBmPKS2Y3XqQe9Xu+MFyCAAALpE1hOp+zNm/dS3NPup5/0\nGqQFukDedELdBpv+11X9R/dvLdtFcauCggACLRWYpA5Vsl/UIbwfOYwtzJb/g9kI/wNMQaDX\nAqpRem2S7jlVD3hbqPndcN2rdIWSofnlG1Ly1KGapYPUscN1Q9QEb4xlfnGK2R7l6zGNAAII\nIJBcgaTXIJVfOU8IKzw7o2S1XTXlidVjJXMbO0ENUmN92XpbCozXwz8DJUXRnuzCy/UZ96i2\nPB0OOnYCJ6nVwEC1LFBS9E39ru3c/QGGz2ctuCKw7BVKtl7ofl2WIoAAAqkTSFQNUtoSJP9t\n9Z6M/L4kPWsl9w31Oxq2ulkbCZIuAgWBZQUmqFfm8ueVZU/V429+uuy6zEGg7wLqqGE7ddww\nMrDwSCVL63S3JdU83aP1rlCXi9eebfZmd+uyDAEEEEiJAAlSG17oHXXMxysOUaxd4fhf1Ly/\nK36keKPC8kbPIkFqtDDbb2OB8RfqA+txpScQ6n7CiVeVzmMKgf4LDNdTitWM4DP6nVOyZAdr\n2E2zzlzHD39TBxD+zK4b1ISv8Ly9/h8IW0AAAQTaS4AEqb2ul+nbZjs9f8x+H9JMhd/o7fc3\neE3SYMXGinUVcxQnKpr9wYsESegUBCoLjNA9IZverGX60Foo/pyb7CfMOu8qzGGIQL0FVH05\neKCek6eOHY5UM7weHgURetPsm9W5w9X6R3LLlCVNtet9SGwPAQQQiKsACVJcr0yF4xqhedcq\nblX8UPGQolLxpob7KNRaItdZg/8jvFvRrEKC1Cxp9tOmAt6zXaBkKNi+eAKhmjZl91CSxP0g\nRRTGGiSgJnhDtekjM0qW9Hu4RXe7URM81SQFN+p+pWvVC8RfLljSs1N3L2EZAggg0O4CJEht\ndAWv1LHqA1Sum+8KT1Zf5kz8/qQZiqsUxy6ztHEzSJAaZ8uWEyNwykZmy92n01kvckrPm3Xp\ny43O1yLzGEWgoQKjzXZVL3dHqGndV5QIRX8fl9mvkqV3PVlSz3nX6p/QX0mWliFiBgIIJEOA\nBKmNruNjOtZHFPrGr+byL635tuLgml/R/xVJkPpvyBZSITBhZ53mHYqViqcb6n0e7Gd2lr9v\nKQg0UyAzzky/e5nDlQh9Sc3wvMl2NyWcq+Z6NytZur5LLRvONVumm/FuXswiBBBAIM4CJEhx\nvjplx/ZXTetbZ9tBsahsWaXJQg3SxVo4ttIKDZpHgtQgWDabRAHv/tuuV1Kke5OWFtUsvftJ\nswv93kIKAk0XGGU2cJDZp5UsfVU1S4coWVKz0O5KOE/PYvqz1r1+gdmfVLOkmiYKAggg0LYC\nJEhtdOm+pmO9QnGT4kyFN8+pVPwepI8rJiv8G2p9I2h3Kfpa/FvEsxT+y1JL2Vwr+ZOHV1F8\nUMsLWAeBdAtMGKkPlr9WkuTv3XwJbzOb91mzC9SSiYJA6wRO0GMk9ByJz6jb8MP0e3pwT8mS\nap/8d/Yf6jr8j0qabvSH2rbu6NkzAggg0CcBEqQ+sbXmRf7hSc8CtDMU3iRnpuIVxRyFf1u3\nmsKTmU0U6ykWK3Qvrp2v6E8hQeqPHq9FoCaB8d9TgnRe6arhH/U9yJfNpvp7mYJAywVGmq2w\nltmBSpbUaVD4uRqSJeVIdq/WvUHN8G44x+zZlp8EB4AAAgj0LECC1LNR7NYYpiPyGqR9FeuX\nHd08Tc9S6INVLjF6uWx5Mya/o51coqAGqRna7CNBAuNPVZJ0eukJhao1nvgNzfMPmhQEYiPg\nNUsrmB2gZOmLqjU6tOd7lvyXOHxa9y3dpB7xbpqu3lWvU68ksTkhDgQBBBAoCpAgFS3acsxr\njdRU3PR/ymYr5ipaXUiQWn0F2H8bC4zXl+zByWUnoIfLnuUPh6YgEEuB4Xog7S5mwz1ZUtO6\nz+t32Fsx9FDCOUqW/qxOHm5Sm7xbuW+pBy4WI4BAMwVIkJqpnZJ9kSCl5EJzmg0RUFPa8b/S\nB8yjy7b+cyVJJ2oeNUllMEzGTiAYY7abkiUlSuHnVbO0VU9HqJol73joTiVXt6hK6U9nmz3d\n02tYjgACCDRQgASpgbhp3TQJUlqvPOddL4GM2YTfaWP+cOhoUdPVs47VDJKkqArjsRZQ1+Fb\n6hdWvTUGSpZsdw31+919UcI0TYnVn7ose0tWN+GdSxfi3YOxFAEE6i1AglRv0Rht7zgdi3+Y\nukjxyyYeFwlSE7HZVVIFRg00W+N/9WHyoNIzDKfoniSvXdLnRgoC7SWg+5bWXsHsICU/B+vI\nP6Xfb79XtduiZMmfr/RPfS9wqxKsWyeZPdPtC1iIAAII9F+ABKn/hrHdwmk6sh8rfHi6olmF\nBKlZ0uwn4QIj9Ad62DX+zXvZiV5p9sJRZtdxg3sZDJPtI+CdPOgXfHjGMuo6PNQXAcGQ2o4+\nnK57l/6ijh5uVeZ0G/cu1abGWggg0CsBEqRecbXXyuvocD1ez0ezjp4EqVnS7CcFAsMHmO1x\nlU60vLmdOgC79wi6AE/Br0BKTvEks61VbeqJkp7/ZR9XLZMmeyrhYjXfu08J099UqfrXGWb3\n643BFwc9sbEcAQR6EiBB6kmI5b0WIEHqNRkvQKA7gREdqkm6XB8clRBFS3iD2YtfUU3Swuhc\nxhFodwHVLq2mpngHmGWULIWfVrK0QW3nFM5VwnSbaqT+roeH/Z3nLtWmxloIILCMAAnSMiTt\nNWMNHa53860Hndv7incUHyhaWUiQWqnPvpMqoBvbx1+mJGlk6QmG+uY8/JLZpPdK5zOFQHIE\n1O/99qpO+oxqij6j+5A+rveBf3ipoYR6mHrwj6xlPWH6x3lmr9bwIlZBAAEESJDa8HdgRx3z\n8YpDFGtXOP4XNe/vih8p3qiwvNGzSJAaLcz20yqgz4YTvMOVUaUA4UNmC/RN+/nenJaCQKIF\n1IX4ynoj7K/apU/py4FPqXZpy9pPOHwqa8Htun/pNvVy8s+zzd6s/bWsiQACKRIgQWqzi32q\njvf0/DG/pOFMxVsKrz3ymqTBio0V6yrmKE5UXKVoZiFBaqY2+0qbgCdJP9NJ/2fpiYfTdA/G\np806nyudzxQCyRb4nv7nqXbpAHX24AnTJ1RjtGYtZ6ze8dQazx7T+rfrYbW3q+nFHReZvV3L\na1kHAQQSL0CC1EaX2G/SvlZxq+KHCn1rXLHoA5Tto9CXY7aLYm/F3YpmFRKkZkmznxQLjPsv\nfYP+k1KAUN+Ge29gk+4vnc8UAqkRyIw221HtUT+hmqVPKgPyzh5WrO3sQ1Uq5RKmf6pJ3lS9\n9k5qmGqTYy0EEihAgtRGF/VKHeseim0UH9Zw3H5/0gzFVQp/HlKzCglSs6TZT8oFxul5SBlv\ncqee7paWeerES1+mdN6ydA4jCKRU4ATdn6sbdPdU7ZKSpfA/9O3hrqoxir5fqsrka5ie0H1P\nd6hJnsLunGQ2q+oLWIAAAkkSIEFqo6uppgD2iOLIXhzzv7SuNxk4uBev6e+qJEj9FeT1CNQs\nMF41RoHXLK8UeYnuR8/qS5FJ6tSBggACBYHvmq2i6qR9Asvsn1HCpPm6pzdQhVOtJXxR69+p\nJnl36BV3dprRpLVWOtZDoL0ESJDa6Hr9Vce6kWIHxaIajrtQg3Sx1h1bw/r1WoUEqV6SbAeB\nmgTG7G7WcbM+uK1Vunp4jroBH8cDZUtVmEKgIKBnL62uT0H7hpYZruap+6mW6GO9SZhUyzRb\n27pL3YrfpW8l7tLNwA9eUtv/Z72MggACMRYgQYrxxSk/tK9pxhWKmxRnKu5TVCr6G+/doNpk\nxc6K/RR3KZpVSJCaJc1+EFgqcMoWZgN1f2IwdOmsJSOaF37VbOLcsvlMIoBAmcAodXa0Wu7/\nZ0bJUjhci3fSPUx6DlltRQnTfP0DfkDvw7t0H9M9uo/pHu5jqs2OtRCImQAJUswuSHeH44mP\nvvCyMxTenGam4hXFHMW7Cv1dz/Vit4mG6yn0hZaNUZyvaGYhQWqmNvtCYKnA2HV1T5K+QAl2\nWTorNxI+oy+1DzE759nS+UwhgEB3AvkmeX4P0776okFN80y1tYGeYVt7UdL0nJKsuz1h6lKH\nSeeaPaFXe4cQFAQQiK8ACVJ8r03VIxumJWcq9Afb1i9bSzdo524i/aOGnhi9XLa8GZMkSM1Q\nZh8IVBQ4WbdYLPdrfYhTrVG0hO/oM9lX1HmDN9WlIIBAHwTU+8lyaue+i6qU9P838B5i91Ly\n44/X6EUJ31fN0v16nVqBZO9VpnSvmnt4Uz0KAgjER4AEKT7Xok9H4rVGgxT+jZb/gY1DMxoS\nJF0ICgKtFRj/A+1ftc2B1zznS6gvsEPVKk86rzCHIQII9EsgGG22td5ke6vjBzVtDzUMNu3t\nFlXLNM0TJnX+cL9ee7+afzykmqb5vd0O6yOAQN0ESJDqRsmGCgIkSAUJhgi0VGD8oUqQ/L7F\nVcoO42q1yh1ldqHuKacggEA9Bb5nto53LZ61jGqXQoXfC9y7ZnlKtLyJ/GPqYlwJU/Z+1Tj9\n3ww1zbtOffjX81jZFgIIVBUgQapKw4K+CpAg9VWO1yFQd4Ex26mHuxv1AW1o6ab9vqSuL6sv\nl8dL5zOFAAL1FPBmeboxeEdtc0/VMu2h5GcP1Rb5vcK9LOE8JUoP67XqBCL7f8qgHjhnSTfj\nmk1BAIE6C5Ag1RmUzZmRIPFbgECsBEavpWfJ6svnYHjZYfk9i8ebnTWlbD6TCCDQQAE9d2Nd\n1SztoVomJUuhkibbRYnPyr3dpZrmeQdND+m9/aCSpgdVvfQgSVNvFVkfgYoCJEgVWZjZHwES\npP7o8VoEGiIwfIA64DpTm9Zns+h9Sb6z8DdmC5Uoncs9Dw2xZ6MIdC+gWqaODc22UecPu6nH\nPPWUF+6maqHtlDRpVu9KPml6WO/zh3RP00NKmh5Wd7dP6xsSmuf1jpK10y1AgpTu69+QsydB\naggrG0WgHgJjP6fPYpdrS2uUbi18TE8GOMzs7KdL5zOFAAKtEBilx3msqucwZcx2Vff9inAX\nHcdmSppU+dTbEnpt8aO6p+lhJU0Pa5sPq1enx6eYLejtllgfgZQIkCCl5EI38zRJkJqpzb4Q\n6LXA6E3U5O5afcO8W9lL9SHKe7mbeFHZfCYRQCAGAnoQ4uqqClZzPFOyFPjQO4AY0pdDU01T\nl17/tCdNgWX/rW38W1VMj5xt9mZftsdrEEiYAAlSwi5oHE6HBCkOV4FjQKBbgRH6479pp1Y5\nscJqt6iH4W/pUWqvV1jGLAQQiJGA2sauuZISJTXJ21m1S4q+J01+WkqcZmkbjyhxUmQfUWcQ\nj8w0e5YmejG66BxKMwRIkJqhnLJ9kCCl7IJzuu0sMO7L+gb6UsWg0rMI/Vvkb6s26Y+l85lC\nAIG4C5xsNnjgkp7zdgots6PSnp2U9Gyu97la1/WlhAuUgD2p1z+qziDUVM/UJNcencwDbvuC\nyWvaQ4AEqT2uU1sdJQlSW10uDhYBb3I38Ldy2G9Zi/Ays/fUsodnJi1rwxwE2kfgu3oe2spm\nH9URf0zdjXvS9DElOt4RhB7b1Lei2qbZSrz8eU2Pq7bpMTXRe0y9SjyhxOmDvm2RVyEQGwES\npNhciuQcCAlScq4lZ5IeAX2zPF73HwU/1Sn7P4ZoeVETel+fdVt0JuMIINDeAsN1M6Iypa0H\nKmlSTZOSp/CjSng0DNbu65kpaVLeZdMVj2s7T+j+JiVP9sQbut9pCp1CiIXSJgIkSG1yodrp\nMEmQ2ulqcawIlAiM/Zh6zLpCH2y2LZmdm8h6UzwlURPnLruMOQggkBQBVRmvp6QplywpcdpB\niZPCtlRtk2b3reQ7hXjBkyVt58msZRX2xFtmz0whceobKq9qpAAJUiN1U7ptEqSUXnhOOykC\nI1cwW+csnY06cCjvUjicpQ9Lx5lNujEpZ8t5IIBAzwIjVLO8kWqb1IveDp406QG32+eb6G3Q\n86u7WyNUnmQvaltPqqneU6px8sTpKTXVe3qS2vd290qWIdBAARKkBuKmddMkSGm98px3wgQm\nfFLJ0K+UJA2pcGLX6LlJSqAmz66wjFkIIJASAX1bsoZ60dteCc12Spy2V+KkYe7eptX7S6Ba\np5fV5E/PZgue0j1OT/u4tv1Up9lr/d02r0egBwESpB6AWNx7ARKk3pvxCgRiKjBG93V3nKkP\nKCfoAMt7wJqjeeN0b9JvNNTnFgoCCCCwROBksw1U27TtkmQps20+cdpazev0/Nv+lnCutuuJ\n0zOeOGn8GdU6PaMuyZ+/wOzD/m6d1yMgARIkfg3qLkCCVHdSNohAqwXG7KlEST3aBVsveyTh\nvZqvTrLOenjZZcxBAAEEigLfM9tYnzy3UVKzjXrT88QpPx6sVlyrb2N+n5NeOV3xjBKxZ5U8\nKewZZUzPKnGaqfmapCBQkwAJUk1MrNQbARKk3mixLgJtI+APlx32Xzrc8UqIBpYdtr7ADS/S\nl7c/MjvvnbJlTCKAAALdCqgqen39EVHSZPoSJqMIt9ELvMbpI92+sOaF4Tyt+pzuc3rWh+ok\nwpOn57S/Z8828+e+URCICpAgRTUYr4sACVJdGNkIAnEVGLODapMuVpK0R4UjVG++uWZ3v9WQ\nb2srADELAQRqF/B7nFZUoqR7nLbSPU5bqcZJYVvrj8tQJU+a3f+imif/Uuc5/U173ofqKOJ5\nNdfz5Ol5kqf++7bpFkiQ2vTCxfmwSZDifHU4NgTqI6DPDuO+pQ8U6u0uWKvCJu9TfnSKugS/\nu8IyZiGAAAL9EvBe9TY220x/iNT9uEdmK/3N2VIbVQRr9GvjJS8O/bEGz6vmyZMndVOefd4T\nJ8ULk8xe1Ty+CCrxSswECVJiLmV8ToQEKT7XgiNBoMECE/yDyH8rRinKO3HwfV+nmKD7k170\nCQoCCCDQaIHRZmvpj9GWyly2UOK0hSdOSmg0tE2VPK1Qr/2r5mm+tutdlL+g7XrylBv36UVm\nM+gwol7SLdnOctqrd/ixl+KelhxBHXeq31NKDARIkGJwETgEBJorMGFnfQi5UB8Sdquw34Vm\nWX1WWHgG9ydV0GEWAgg0SyCj5GkjfVjcXAmUEqbM5kpyPHHaXDFETfYG1u9AwqwSpZna3ova\nrhKnXPLkXxRN0/6n0VV5/aQbtCUSpAbBpnmzJEhpvvqce5oF9H8/1+xOiVCwbgUIdQue/Yk+\nG/xSFUtKmigIIIBAPASGmw3Y1WwTdRThzfY2V83TZvrSxxOnTTU9VH/T/ANz3YrXPmljuWRJ\n256mTiOmKaGarpimBdPV483bddsZG+qLAAlSX9R4TbcCJEjd8rAQgaQLfHcVs9XG6Sz1Za3p\nGZLLlBlKlE7X54DLlSh5t7wUBBBAIM4CmZPUPbme66SkyROmXPKk5nq55GmYEhz9zat3yd37\nlEuYtP0ZqoGarlovn57+vprvkUDV23uZ7ZEgLUPCjP4KkCD1V5DXI5AIgXEb6h/7mTqVr2uo\nL2HLS/iMEqUfm3VeqyX6v09BAAEE2k9Az3ZaR93pbaoERslSxmuchqm3vWH6o6ZpW09N7Cr8\n/evfeaoG6l1tVImTTdff15f0t1RDm6He92aoFmzG+WazNc3fVSH0sZAg9RGOl1UXIEGqbsMS\nBFIoMHYnfWhQb7nB8MonHz6if+4/UqJ0c+XlzEUAAQTaU2Ck2Qqrq4meap+UNOUSqKHKWxS5\n5Mm7Kl+1EWemBOpD7e9lbXvGkgTKXlIzPo3by0qgXlLm9PK5Zt7Mj1JZgASpsgtz+yFAgtQP\nPF6KQHIFxn5Gz086Q+enDh0qlfD/NFfLJ96oId98ViJiHgIIJErgeLM1V1YCpbbGQ1QLNVTP\nelLSFA7RH0BPojZRAqXHQDWqhG94oqStqwYq0DCrob2sY/F5L+vHLHVDmtZm0CRIjfq1S/F2\nSZBSfPE5dQR6ENCXmhO+qHV+otim8rrhY/pHraZ5nfrfbPqyk4IAAgikU8Cb7w3M9bBnQySw\niWrj1dteLoHSeC6BUn7VmKJaqC79wfZnPb2i50B50vSK/iTnhp5EqSvzV17T8oQmUSRIjfm1\nSvVWSZBSffk5eQRqEsiox7uv6Z/9aVp7WOVX+D1K/oyl+64ym6qm9RQEEEAAgaiAesLxB3Xn\nkiX9UfUEShFunJ+3sWqgKj3IO7qJfo17EqUNKE/y5MkjeCWw7MxQ4/p2a6aOaeYbiilmC7S8\nnQoJUjtdrTY5VhKkNrlQHCYCrRcYpS9HB39T/9D1MNlgaOXjCadr/nlm711mdqE6cKIggAAC\nCNQiMEo9ieomp42VqHjSpIQpkxsqsfHnQW2sREbDYPlattW/dcI52tdM7VPJUzBT25rliZQn\nUUvm2azJZsqlYtO8mgSpfxecV1cQIEGqgMIsBBDoTmD4ALPdVKMUfF+xZZU19VyQ8GI93Pxn\nype82QcFAQQQQKB/AoGa8X1EGdJGSlY20qaUMGXyw3CjJQmUrae/y/ob3diipE2t9uw1T5g0\nnKV9zlLHErN0DBr3hMpeVXXVLHUu8VZjjyS3dRKkJiCnbRckSGm74pwvAvUT0BedY0eomcgP\n9c9x+yqb1UNmw6t077B6xpv8eJV1mI0AAgggUAcB/UHu0DMb1u0w2zCfMG2YWZJEaTrcQLvQ\nYltfNVED67C7HjehfXoPfY93WXiM/gk82OML+rYCCVLf3HhVNwIkSN3gsAgBBGoS0P+/8Yco\nSRqvtfes/orwH0qWfq6Hzt7EQ2erK7EEAQQQaLBAZqxqovSH25MlT6Q2UE3UBqqV2lCdSngS\npWnTsJ4P1Q3/NcnCfRp0XiRIDYJN82ZJkNJ89Tl3BOouMH4vbXKM/rEeqqFqmCqWGZqrh8u/\nd6nZL+ZUXIOZCCCAAAItFTjBbDXVRG2g9nqeMK1fSKT0Rdf6OjDVQnkSZevq732PTfpUkzS1\n08L9G3RCJEgNgk3zZkmQ0nz1OXcEGiYwdnPlRydr8yP1z7Pas0HUU1J4teJCs0n+XCUKAggg\ngEB7CWR0X9TaSqTW1zdiucRJf/s9edK9ULlESkN7M2vhyWpi93SDTo0EqUGwad4sCVKarz7n\njkDDBUavZTbgWCVJity3jVX2GD6sf6aX6GHxV5ld8G6VlZiNAAIIIIBAuQAJUrkI0/0WIEHq\nNyEbQACBngW857s9vqD11GrDummHHn6g5deoUwclS5Pv63m7rIEAAgggkHIBEqSU/wI04vRJ\nkBqhyjYRQKAbgfEfVW3Rf6pW6WvdNL/z13uvd1NUq3SF2fmv+wwKAggggAACZQIkSGUgTPZf\ngASp/4ZsAQEE+iQwYQ0lSt/QS/V3KNi2m00s1np/0Tq/NfvgRjXB+7CbdVmEAAIIIJAuARKk\ndF3vppwtCVJTmNkJAgh0L5Dr/c4TpcO03krdrPu2kqXfablqlSbeo6E6VqIggAACCKRYgAQp\nxRe/UadOgtQoWbaLAAJ9EBg/SC86UnG0kqUde9jADOVH6gWvSzH50R7WZTECCCCAQDIFSJCS\neV1belYkSC3lZ+cIIFBdYPz2WjZSiZLuVbJ1qq/nS8In9OMqs6w6eOh8oft1WYoAAgggkCAB\nEqQEXcy4nAoJUlyuBMeBAAJVBLwHvN0/rYUjFQcrYVq+yor52d5luP3ebJHinGe7X5elCCCA\nAAJtLkCC1OYXMI6HT4IUx6vCMSGAQBUB79jBvqQaoyOUKO2ncT2bsLsSPqZ1r9MaSpYmPdXd\nmixDAAEEEGhLARKktrxs8T5oEqR4Xx+ODgEEqgqMW1/50Ve0+HDFrlVXKy5QbVL2j0qYFJ33\naHa2uIgxBBBAAIE2FSBBatMLF+fDJkGK89Xh2BBAoEaBkzczG6hkKfiy4mM9vyicrXVuUihZ\nWvh3s3Pn9/wa1kAAAQQQiKEACVIML0q7HxIJUrtfQY4fAQTKBDxZWs4TJYXtXLawwmSo5Ci4\nXRVKf1KN1C1mZ02vsBKzEEAAAQTiKUCCFM/r0tZHRYLU1pePg0cAge4Fxgw169A9S3aokqC9\nNOzhnqXc1p4sJktv3WV2yaLu98FSBBBAAIEWCpAgtRA/qbsmQUrqleW8EECgTOCEtc1WVi94\nniyFByhhWrFshUqT72vmVCVMf9FrFJ3PVVqJeQgggAACLRMgQWoZfXJ3TIKU3GvLmSGAQFWB\nUSuZrf4pJUnebfiBWm29qquWLAina/2/apbuW1qkZnlnv1mymAkEEEAAgWYLkCA1WzwF+yNB\nSsFF5hQRQKBbgcBs7I5KfA5SCzxFrke8GprihaHWfURxm2qY/mH2wR1mF3qNEwUBBBBAoHkC\nJEjNs07NnkiQUnOpOVEEEKhNwJvireQPplUEaopn69T2Olus9R5QU7x/KmFSBHfp2Uvv1fha\nVkMAAQQQ6JsACVLf3HhVNwIkSN3gsAgBBFIv4LVLH1VHD0qWQm+S93GJ+D/jGkrYpfUfKiZM\ni+9Wd+Jv1fBCVkEAAQQQqF2ABKl2K9asUYAEqUYoVkMAAQTMxqysZElJUvgJJT8e/sylGprj\nuV2uSZ56yPOapey/lsTkaagigAACCPRLgASpX3y8uJIACVIlFeYhgAACNQmcPFgPqB2uHEnJ\nUri/kp+ta3rZ0pXCWRq9R69TdN1r9saDZlMWLF3MCAIIIIBATwIkSD0JsbzXAiRIvSbjBQgg\ngEA1gTEfMRuwn5IlReDDbTVUM71aS+jPXHpYr1Gy5AmT3a+uxV+o9dWshwACCKRQgAQphRe9\n0adMgtRoYbaPAAIpFjh+TbNV91Wi5PcueXhveQN7BxL6fUvq/CFQshRquEDD81/v3TZYGwEE\nEEisAAlSYi9t606MBKl19uwZAQRSJ+DPXxq8WyRh2kOJz6A+MMzUa9QcL1T4MKvofK0P2+El\nCCCAQLsLkCC1+xWM4fGTIMXwonBICCCQGgE1vxut+5Y69lSipDBPmPw+pho7fog6+f1M3mue\nN9HL/luhIZ1ARIUYRwCBRAqQICXysrb2pEiQWuvP3hFAAIEygfGqUQpVy2SKjA93VaxXtlKN\nk+FcraiH2QZKmHyYfdRs0RPqbnx+jRtgNQQQQCDuAiRIcb9CbXh8JEhteNE4ZAQQSJvAuA11\nxkqWAiVLwS4a31mxRh8VskrAntN2lCyFCnvMbKEnTS9qXMsoCCCAQFsJkCC11eVqj4MlQWqP\n68RRIoAAAmUCY4aqhkmJUuDJkg930nDNspV6MzlPKz+leFyJk8KUNC1+0uzslzQeKigIIIBA\nHAVIkOJ4Vdr8mEiQ2vwCcvgIIIBAUeCUjdTN+I5KlvwBthqahsEQDftRwg+0jae1ASVLYT4W\na/oh1ThNXdyPDfNSBBBAoB4CJEj1UGQbJQIkSCUcTCCAAAJJEzhpdbPld1ByowjyEW6n8ZX7\nd6a5ZzY9r208o1DCFHoSpfFFz6q53lv92zavRgABBGoWIEGqmYoVaxUgQapVivUQQACB5Aio\nl7yTh5kN9IRJyZL5A219uIViQB1Oc462oUTJIyyEkqkPFBe+X4ftswkEEECgIECCVJBgWDcB\nEqS6UbIhBBBAoN0FRuiDxrAt1VeDkqXMNjobD3U7HmyuYT0SJ20m9Oc1ec2TIsxHVs31si9Q\n8yQVCgII9FaABKm3YqzfowAJUo9ErIAAAgikXWDUQLNVlSR1FBKmrZQ0KZEyj1XqpxO+o20p\nWQqULPnQIztNzfYUr6iziOsW1m9fbAkBBBIiQIKUkAsZp9MgQYrT1eBYEEAAgbYT8C7IM0qY\nwkLCtIVOQRFsoqGa8tWtZLWlmdqPkiWbviSCGcXxt142u2SRpikIIJAuARKkdF3vppwtCVJT\nmNkJAgggkDaBE5Y3W25T1Tp5srSZQjVQoYbm40qq6po8Oa4SqHCWhqpp8vDkKathmI/FGp7n\nNVQUBBBIlgAJUrKuZyzOhgQpFpeBg0AAAQTSJDByBT2yaZiSJ0+YNMwokQoVPh7o+U7mH3ga\nUdRBRKiapkDhQ4+MD1UzZa8siYlzG7FjtokAAg0TIEFqGG16N0yClN5rz5kjgAACcRTImI1X\nDVNWiVLgyZOGoY/nh7aexoMGHrj3sqdkKZc0KXEKFD4eqHYqq+EixcOv8wyoBl4BNo1A7wRI\nkHrnxdo1CHxH61yi8Jts9TBACgIIIIAAAnEW8KZ7K2yshGVIPvxeJ43nQvPD9TW/Q9ONLH4/\n1Bval5ImU4SvLkmgckONd6mnvg4NX9SQjiUaeSHYNgISIEHi16DuAiRIdSdlgwgggAACrRMY\nPsBs5w1U86RkKePJk4YewUZKZBSmCPTw3GaV0B+aq2TJlCyZap58GChCjWc1zPhQMV0J13Vd\nWk5BAIHeCZAg9c6LtWsQIEGqAYlVEEAAAQSSJPBdtZpYJZ8wdag5X+idRkQi14nEGs094zDU\n/vwBu55EecxWIqXI5oc+3eWhRGqxhhe8q3UoCCBADRK/Aw0QIEFqACqbRAABBBBod4GTV9Sz\ncdVcL6PaqKzXSClM06ZhoPugvCmfTwdaryVlofbqzfwUPlwab2qeIqN5XT5ULFA8ouRr6uKW\nHCk7RaCxAtQgNdY3lVsnQUrlZeekEUAAAQTqIzB+UCRZUuJk6yq8IwlPojS+dNjEZn3Vziz0\nHvqULHlNVZAfhl5rFQmf9sioaeB7igu90woKAnEWIEGK89Vp02MjQWrTC8dhI4AAAgi0k4B3\nLrHSOkuSJk+igvy4D83jI/mhxpt5j1SPhqqp8vuoAkVhaG8vGQ819Pk+nfXlPlQsUnyoZ07x\n4N4edVmhHgIkSPVQZBslAiRIJRxMIIAAAggg0GqBEfrAN1QJU+hJkyJYWzU6kWnTdCF8Wa4n\n2lYfdKX9e+2TkqXQH9CrYaBhdDyr6YyiMDTVcC3W9ADFC7rHik4rKqEybxkBEqRlSNprht/w\nOUihb5HM/2joD0DLu9YmQdJFoCCAAAIIINC+Av7g3cFKlDo8WVpLSYcil0BpGChCn15zyTKf\nzo37h8qYl/A9HaA3C4xELsnyDiry88LIeCY/3qXhAh9XXPChhpRkCyQqQVI3nKkoO+osj1cc\novA/XOXlRc34u+JHCr/JkoIAAggggAACCPRCYMoCrfxyPmp83bhVlUh50qToyg8zg5dMB/np\nUNOBojC0NbRcD/JtVgl0jObhPQxGSvQ5wdHxwir+GKyV8xMTvImgEqVAkUumNLT3ItMaz83X\nMDdfw2x+nVyC5ssVc943yznnt8sAgcYIVPqNbsyeWrfVU7Xr0/O7f0nDmQpvq6s3Wa4mabCG\n/nwGv6FTN0TaiYqrFM0s1CA1U5t9IYAAAggg0L4C+ux2klrCrOCfX5QsZTUMNPQEKpc8eQKl\n8HnRaVtd81bTvCYmV9pb/Yv3AphPmAL/LOfjPsyP+zyf9sjkh76OtxbyeVlFhyLQ9CINSbpk\nU4+SqBqkpCdII3TFr1Xcqvih4iFFpeIO+yjOVuyi2Ftxt6JZhQSpWdLsBwEEEEAAgfQKKDk6\nSUnSCkqWcsmVhhlFNj/0L45DjXsHFT7MfZHstyXkx3MJVgJbH4V6OLAnTLlbLjyRUuSSr/x4\nYb4PC+tlNZ6Zp5o/DX1+h8YX58cH+HzFAsVrmpeK+7hIkPSL0C7lSh3oHoptFLW0f/VvW2Yo\nvAbpWEWzCglSs6TZDwIIIIAAAgj0Q2CM2s1lPGnKR6iEqyMynpvvNVWF0DJPrHy9XILlzfVW\nUbR7TZZOoebiTQyVLHly5UPzcUVh3OaXzgs07ctyr8mP+zq59TX05VnfRn48o+FripY2P0xU\ngpTAbwFKfll30NQ9ilqSI3/h24pHFf4gOgoCCCCAAAIIIIBAicDkfC2JzSqZ3bsJtdwZpyRp\nkZKmjBKmDg1z9zl58qRYmkjlpwvLCvdDhZ5g+bL8MPCOt+JclDwEnkB47VzkOKPjhdmFeT6M\njheWF4bl+aXfKTI+1I+sXveOkilPnhYsGXpyFbwg7++rsdSMwhYYVhdIeoL0qk59Z8VAxaLq\nDEuXeA2SJ1UXL53DCAIIIIAAAggggEA9BfRBfpLfF+RRhzJcn2f3UMK0UAnTAEWXokMRKHIJ\nl4aeVIWq/fJ5ufkazyVavo73JlEY+nh+OvCeJtqoBJ5R+TGvqXPMH3dhaLvp47B/zj0wv4BB\nNwJJT5B+q3O/QnG94kzFfYpKxX97Pq6YrFhJcYOCggACCCCAAAIIIBB7ganquGGqtwLyqGPx\nBwsPULIU5GOghp5keQKVS7b8M2N+Ovf5UeOB5hWWhYXlGvp8Xyc3rzDttUpNLLmOPJq4v/bd\nVdITpKt0aT6iOENxsGKm4hXFHMW7Cq/S9V5fNlGsp/CeUUYr7lJQEEAAAQQQQAABBFIrkHt+\nk9+m8VZjCEaotmeokqUPFQMVHYpwxSXJVFbjuaRK0xmFT/vQEyxfJzeuYS4h09CGanxDLfPm\ndN7Ezuf58vww1Gfg7JjGnAdbbVeBYTrwqxWeIHn7zGh4W9rnFF57tJGiFeU72qkfk38LQUEA\nAQQQQAABBBBAoJ0EvDbMP8vu2U4HXe1Yk16DVDhvfxDs4fkJrzVSjyqmJ17bbMVcBQUBBBBA\nAAEEEEAAAQQQSFUXi4XL7TeveXj3H4Ub8grLGCKAAAIIIIAAAggggECKBcr7CEwqxY46sUsV\nXmPk7UinKZ5W+P1IeoqyqevDXM91a2tIQQABBBBAAAEEEEAAgZQKpKGJ3am6tqfnr+9LGt6j\n8CTJEyNvauedNGysGKX4kuJEhXfuQEEAAQQQQAABBBBAAAEEEiUwQmfjN4z9WbFTN2fm3Xzv\nq3hA4evvpWhmoZOGZmqzLwQQQAABBBBAAIF6CiSqk4Z6wsRxW1fqoLz5nPqxr6n4A7S8++9f\n1rR2/VYiQaqfJVtCAAEEEEAAAQQQaK5AohKkpN+DtIN+N7xJnfdhX0vxB4w9qtiglpVZBwEE\nEEAAAQQQQAABBJIlkPQE6VVdrp0VA2u8bF6D5EmVd+BAQQABBBBAAAEEEEAAAQQSJfA1nY3f\nU3SjYvduzszvQdpHcZ9isWJvRX/KML3Ya6P8IbS1xAKt58fpTzumIIAAAggggAACCCDQTgKJ\namKX9F7svDe6jyjOUBysmKnwrr3nKPxeo9UUgxWbKNZTeHI0WnGXoj9lul58mKLWmqutte5k\nRZeCggACCCCAAAIIIIAAAgg0VMBrdK5WeILkNTXR8Bqe5xSeoGykaEXZUzv1Y/Lsm4IAAggg\ngAACCCCAQDsJUIPUTlcrf6wvanh4ftxrjfz5Ryso/MGxcxUUBBBAAAEEEEAAAQQQQMCS3sSu\n0iX2pnUeFAQQQAABBBBAAAEEEECgRCDpvdiVnCwTCCCAAAIIIIAAAggggEB3AiRIpTrHafIR\nxbGls5lCAAEEEEAAAQQQQACBNAiQIJVe5XU06c9B8iEFAQQQQAABBBBAAAEEUiaQxnuQurvE\nF2nhHxSvd7cSyxBAAAEEEEAAAQQQQCCZAiRIpdfVEyOSo1ITphBAAAEEEEAAAQQQSI1AGhOk\nNXR1vZvv5RXvK95R+LOQKAgggAACCCCAAAIIIJBygbTcg7SjrvOlCn/u0VuKaYqnFa8oPEl6\nQXGxYm0FBQEEEEAAAQQQQAABBBBIrMCpOrMwHzM0vFtxs+J3ij8r7lO8qvB13lQcoWh22VM7\n9P37U4gpCCCAAAIIIIAAAgi0k4B/hvXPsv6ZlhJzgRE6Pr9Yngjt1M2xBlq2r+IBha+/l6KZ\nhQSpmdrsCwEEEEAAAQQQQKCeAiRI9dRs8Lau1Pa9+Zzfb1RL8fuT3lX8spaV67gOCVIdMdkU\nAggggAACCCCAQFMFEpUgJf0eJH+m0T2KD2v8FXlb6z2q2KDG9VkNAQQQQAABBBBAAAEEEiSQ\n9ATJ7y3aWTGwxmvmNUieVHkHDhQEEEAAAQQQQAABBBBImUDSE6Tf6npupbhesXs319bvQdpH\ncatiJcUNCgoCCCCAAAIIIIAAAgikTCDpz0G6StfzI4ozFAcrZiq8a+85Cr/XaDXFYMUmivUU\nixWjFXcpKAgggAACCCCAAAIIIIBAIgWG6ayuVniC5L3URcMfEvucYrJiI0UrCp00tEKdfSKA\nAAIIIIAAAgjUQyBRnTQkvQapcMFf1Mjh+QmvNRqkWEHhD46dq6AggAACCCCAAAIIIIAAApaW\nBCl6qb1pnQcFAQQQQAABBBBAAAEEECgRSGOCVAIQswmvnmx0qbVHv0YfB9tHAAEEEEAAAQQQ\naI7AogbvphmfYRt8CsXNkyAVLVo5Vvilfa+VB8G+EUAAAQQQQAABBBDoh8DCfrw2Ni/17q0p\n8RDYRYfR6Nqd07SPVRRTFJR4C1ygw7tY8Xi8DzP1R7edBI5RnJB6ifgDjMwf4pT4H2rqj5C/\nf+3xK8Dfv/a4Tn6UIxXvK05TNLJ4cvRgI3fAthFohMAUbdSDEn8B713xs/E/zNQfoV8jv1aU\n+AtM0SF6UOIvwN+/+F8jP0L+/rXHdfKjnJIPH6fUIJD0B8XWQMAqCCCAAAIIIIAAAggggMAS\nARIkfhMQQAABBBBAAAEEEEAAgbwACRK/CggggAACCCCAAAIIIIBAXoAEiV8FBBBAAAEEEEAA\nAQQQQCAvQILErwICCCCAAAIIIIAAAgggkBcgQeJXAQEEEEAAAQQQQAABBBDIC5Ag8auAAAII\nIIAAAggggAACCOQFSJD4VUAAAQQQQAABBBBAAAEE8gIkSPwqIIAAAggggAACCCCAAAJ5gQFI\npEpgYarOtr1P1q/VovY+hVQcvV8j3lftcam5Tu1xnfwo+fvXHteKv3/tcZ0K76n2OVqOFIEm\nCwzW/jwo8RcYpkOkhjf+18mvkV8rSvwF+PsX/2tUOEL+/hUk4j3k71+8r0/06Pj7F9VgHAEE\nEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAAB\nBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAA\nAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBA\nAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQQAABBBBAAAEEEEAAAQQQ\nQAABBBBAAAEEEEAAAQQQQAABBBBAAAEE4iTQEaeD4VgaJuDXeU/FborFircUlPgJrKZDWlex\neoUYqHnzFJTWCnxeu/f30xtVDoP3WhWYJs8eov0dpHisyn55r1WBaeLsYdqX/1/aJr/POd3s\ne0Mt20/hw9mKRQpKcwRW0m52Uuyt8P9N7yo+VJQX3lPlIs2f3lK73FcxSOHvk6yiWuF/VTUZ\n5qdGYHOd6VOKMBJPaHwjBSVeAhfqcKLXKTp+VbwONZVH85389Rld5ex5r1WBafJs/6D2pOK9\nbvbLe60bnAYv8i+BblBE/775+G0KT5rKy+ma4QlRYX3/km9c+UpMN0TgG9rq64qCvQ89QTpR\nUV54T5WLNG96sHZ1oyJ6neZpelSVQ+B/VRUYZqdHINCp3qHwP2hHKjZT+Ic8f+PMUKysoMRH\n4G4din+oO7dC+PWjtE7gUO16ocL/AVVKkHivte7aRPe8hiZuVfh16i5B4r0WVWveeEa7mqrw\n63ON4kDFforLFP5t9+OKFRSFcoBGfN0/KHZU7KYoXN8TNE5pnIDb+zWZpvi+YjuFJ0ZPK/ya\nfF0RLbynohrNHf+rdufX5BKFv0f8/9WdCp93tCJa+F8V1WA8tQLH6cz9DXJMmYAnSZXml63G\nZBMF/IPD+4rbm7hPdtWzwJpa5QqFv18W5IeVEiTea8JpcfmC9j9L4dfKmwBVS5B4rwmnRWU/\n7devj3+YLi9/0gxfNiK/wJt2TVO8ovDmQIWynEZ8/suK6PzCcob1Ebhdm/Hr8amyze2an+8t\nUQqF91RBovnDXbRLv04PlO16qKY9wb2rbD7/q8pAmEynwH06bf9Q5+2Go8WboMxXlL+housw\n3lwBbzvsf+Q6m7tb9taDgL+H/Lpcq/DmJj5eKUHivSaYFpYDtW+/Nm8qDlE8pKiWIPFeE06L\nylHa7zTFtyvs/6ua59fwx/llhWt6VoV1z8yve1CFZczqv4AnPPcrPAmqlIR6LZI3dSws4z0l\njBaVbbTfnyg+WWH/L2jeW2Xz+V9VBsJk+gQG6pT9W9RHq5z6w5rvTYZ8PUrrBb6iQ/APB/4h\nYS+FNx85SuH/eCitE7hQuy784/EP3n6NyhMk3mutuz6FPR+gkZ8qvC2+l+4SJN5rS4zi9vMH\nOiB/fxWaE/84P/3FCgfqTYh8XV+H0lwBbwI5V/F8ZLe8pyIYMRn1Jqldiusix8P/qggGo+kV\n+IhO3f+B3F6F4B/55etXWc7s5gr8v/z1eDY/9Gvn4X/gzlEMUFBaK1AtQeK91trrUmnv3SVI\nvNcqibV23lra/RsK/+C9bv5Q/MsJ/xu4X346Otgnv+yS6EzGmyLw47z9xMjeeE9FMFo46vcX\njVRcrfB7z/0L8iGKQuF/VUGihqFXo1KSKbBa/rS8yUml8lZ+5sqVFjKv6QL+bY+X1xSfVWyU\nHz6l4cmKCQpKPAV4r8XzulQ7Kt5r1WRaM9//B92s8CTpFIX/DfTS3fuK/19LjJr98zDt8FTF\nc4rTFIXCe6og0drhetr9bxTeEmVVxY2KmYpC6e495evwvipIMUy0wIY6O//27foqZ/mH/PJh\nVZYzu7kC/o3otxTefCFa/NvUdxQLFCSzUZnmjx+iXfp7qryJHe+15l+Lnvb4kFaodg8S77We\n9Jq33JMi77DB31fnl+320vz87cvm++QO+WWXV1jGrMYIjNRmFyo8gd1aES28p6IarRtfUbv2\nL1d3UfxS4d3j+31kqyi88L9qiQM/Uy7gTbK8B5PbqzhM1Xz/p7RmleXMjo/AdToUv1beexCl\ndQLVEiTea627JtX23F2CVO01Pp/3Wnc69V22qTbnNRH+t+2MCpv+SX7ZfhWWDc8vu6DCMmbV\nX8Brjfw6vajYopeb5z3VS7A6rl6w/1J+m/yv6gVuphfrsmp7CSzW4c5WDK5y2D5/nsJrJyjx\nFngjf3irxfswU3t0vNeSc+l5rzXnWvozde5UDFGMUvxIUV5m5WdU+h9WmBdtPlT+eqb7L+D3\ntHjN3umKBxR7Kvw+2d4U3lO90arvupflN3dQfsj/ql74kiD1AqsNV/X7V7wLSG/GEC1ra8Kr\nyB9UdEUXMN4SAW8r7NfCm5pUek9ulT+qZ/JDBvET4L0Wv2tS6Yh4r1VSae48b/7zT4U3+/EP\nbr9SVCr+nvJSqQapMO/+JavwswEC/r/o14oTFTcohiteV5QX3lPlIs2dHqvdva34jwq79VZE\nXt5fMsj95H9VBIPR9Ap8Uafu1eLjyggm5Od/uWw+k60TeCx/Tfwm2GjZWxP+R+4f0ZmMt0Sg\nWhM7Pxjeay25JFV32l0TO95rVdkavsDvkZim8HsqvTaip/KoVnhVEa09H6Rpvw/GH1XhTYYo\njRE4Tpv1zw9/UHT0sAveUz0ANXDxwdq2X6f/rbCPP+WXHRpZxv+qCAaj6RXwb4CeVHgt0U8V\nn1R4W2+f9j96lPgIfEKH4tfFex08W+HXyhPb9xRzFDsoKK0VOES7939EoyscBu+1CigtnPWQ\n9u3vnUqF91ollebM+4l24+8hbxrntRKV4tuaXyiHa8TXf1DhX+iNUPi19aZCOykojRFYU5t9\nW+H2/uVcpevk8wo3//OeEkaLijeDvEXh1+qviiMUn1fcqvB51yqihf9VUQ3GUy2wls7+zwqv\nhfA3i8dfFOsqKPES+KwOx9t3F66Tfwi4UzFUQWm9QHcJkh8d77XWX6PCEXSXIPk6vNcKUs0d\neq1P4e9btWF5b3Zf02veirzOx49WUBon4DUO1a5PdP4akUPgPRXBaPLoatrfzxT+maFwfT7Q\n+I8U/nDY8sL/qnIRplMtsKrOfmcFiVH8fw3W0yH6t6Mrxf9QOcIKArzXKqDEdBbvtZhemLLD\n8m/JN1Nsq1i+bBmT8RLgPdW66+FNWD+m8J4GO2o4DP5X1YDEKggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIBAGgU60njSnDMCCCCAQOIFhusMd1U8qdhJcbji\n84q1Fc8puhQUBBBAAAEEEEAAAQQQQCAVAn/QWc5SnKLIKhYpwnw8pOFgBQUBBBBAAAEEEEAA\nAQQQSIWAJ0ieGM1WHKtYQ7Gu4maFJ0rfV1AQQAABBBBAAAEEEEAAgVQIeILkidAxZWe7TX6+\nL6cggAACCCCwjEBmmTnMQAABBBBAIDkCd5edyvT89Gpl85lEAAEEEEAgJ0CCxC8CAggggECS\nBV4rO7kF+Wn+/5XBMIkAAgggsESAfxD8JiCAAAIIJFnA70OiIIAAAgggULMACVLNVKyIAAII\nIIAAAggggAACSRcgQUr6Feb8EEAAAQQQQAABBBBAoGYBEqSaqVgRAQQQQAABBBBAAAEEki5A\ngpT0K8z5Ib9tcuUAAAILSURBVIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCA\nAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggg\ngAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAII\nIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAAC\nCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAA\nAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggAACCCCAAAIIIIAAAggggMD/B/8i\nN8u+ZDWPAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta = 1\n",
    "ecm_MV = function(n) (2 * theta^2) / ((n+2) * (n+1))\n",
    "ecm_M = function(n) theta^2 / (3 * n)\n",
    "\n",
    "curve(ecm_MV, col='darkblue', lwd=3, xlab='n', ylab='ECM', xlim=c(0, 30), ylim=c(0, 0.1))\n",
    "curve(ecm_M, col='darkred', lwd=3, add=TRUE)\n",
    "legend(x=22, y=0.1, legend=c('ECM_M', 'ECM_MV'), col=c('darkred', 'darkblue'), lwd=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae2e2fc5-36b0-4963-9726-2a800a23720d",
   "metadata": {},
   "source": [
    "## Ejercicio 15\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria con distribución de Bernoulli de parámetro $p$ y consideremos el nuevo parámetro $\\theta = p(1-p)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2d8ca19-db48-48ac-a1b0-9d68124364b6",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Proponer un estimador de $\\theta$ basado en el estimador de MV de $p$.\n",
    "\n",
    "\\begin{align*}\n",
    "f_X(k) = p^k (1-p)^{1-k} \\; k \\in \\{0, 1\\}\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "L(p)\n",
    "= f_X(x_i, \\dots, x_n)\n",
    "= \\prod_{i=1}^n f_X(x_i)\n",
    "= \\prod_{i=1}^n p^{x_i} (1-p)^{1-x_i}\n",
    "= p^{\\sum_{i=1}^n x_i} (1-p)^{\\sum_{i=1}^n 1 - x_i}\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "ln(L(p)) = \\big( \\sum_{i=1}^n x_i \\big) ln(p) + \\big( \\sum_{i=1}^n 1 - x_i \\big) ln(1-p)\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "\\tfrac{\\partial}{\\partial p} ln(L(p)) &= \\tfrac{1}{p} \\big( \\sum_{i=1}^n x_i \\big) - \\tfrac{1}{1-p} \\big( \\sum_{i=1}^n 1 - x_i \\big) = 0 \\\\\n",
    "&\\iff \\tfrac{1}{p} \\big( \\sum_{i=1}^n x_i \\big) = \\tfrac{1}{1-p} \\big( \\sum_{i=1}^n 1 - x_i \\big) \\\\\n",
    "&\\iff \\tfrac{1}{p} \\big( \\sum_{i=1}^n x_i \\big) = \\tfrac{1}{1-p} \\big( n - \\sum_{i=1}^n x_i \\big) \\\\\n",
    "&\\iff \\tfrac{1-p}{p} = \\frac{n - \\sum_{i=1}^n x_i}{\\sum_{i=1}^n x_i} \\\\\n",
    "&\\iff \\tfrac{1}{p} - 1 = \\frac{n}{\\sum_{i=1}^n x_i} - 1 \\\\\n",
    "&\\iff \\tfrac{1}{p} = \\frac{n}{\\sum_{i=1}^n x_i} \\\\\n",
    "&\\iff p = \\frac{\\sum_{i=1}^n x_i}{n} \\\\\n",
    "&\\iff p = \\bar{X}\n",
    "\\end{align*}\n",
    "\n",
    "\\begin{align*}\n",
    "\\hat{p}_{MV} = \\bar{X} \\implies \\hat{\\theta} = \\hat{p}(1 - \\hat{p}) = \\bar{X}(1 - \\bar{X})\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51c7bcac-9b85-458e-a3d8-29c5abd28f00",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Mostrar que el estimador de $\\theta$ propuesto en a) es sesgado pero asintóticamente insesgado.\n",
    "\n",
    "Primero hacemos algunos cálculos auxiliares, recordando que todas las $X_i$ son Bernoulli.\n",
    "\n",
    "$\n",
    "E(\\bar{X})\n",
    "= E(\\tfrac{\\sum_{i=1}^n X_i}{n})\n",
    "= \\tfrac{\\sum_{i=1}^n E(X_i)}{n}\n",
    "= \\tfrac{\\sum_{i=1}^n p}{n}\n",
    "= \\tfrac{np}{n}\n",
    "= p\n",
    "$\n",
    "\n",
    "$\n",
    "V(\\bar{X})\n",
    "= V(\\tfrac{\\sum_{i=1}^n X_i}{n})\n",
    "= \\tfrac{\\sum_{i=1}^n V(X_i)}{n^2}\n",
    "= \\tfrac{\\sum_{i=1}^n p(1-p)}{n^2}\n",
    "= \\tfrac{np(1-p)}{n^2}\n",
    "= \\tfrac{p(1-p)}{n}\n",
    "$\n",
    "\n",
    "Ahora sí, veamos lo que nos piden sobre el estimador de $\\theta$:\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta})\n",
    "&= E(\\bar{X}(1 - \\bar{X})) \\\\\n",
    "&= E(\\bar{X} - \\bar{X}^2) \\\\\n",
    "&= E(\\bar{X}) - E(\\bar{X}^2) \\\\\n",
    "&= E(\\bar{X}) - V(\\bar{X}) - E(\\bar{X})^2 \\\\\n",
    "&= p - \\tfrac{p(1-p)}{n} - p^2 \\\\\n",
    "&= p - p^2 - \\tfrac{p(1-p)}{n} \\\\\n",
    "&= p(1-p) - \\tfrac{p(1-p)}{n} \\\\\n",
    "&= \\theta - \\tfrac{\\theta}{n}\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "Podemos ver entonces que $\\hat{\\theta}$ es sesgado pues su esperanza no es igual al parámetro que estima.\n",
    "\n",
    "No obstante, es asintóticamente insesgado pues:\n",
    "$\n",
    "E(\\hat{\\theta})\n",
    "= \\theta - \\tfrac{\\theta}{n}\n",
    "\\xrightarrow{n \\rightarrow \\infty} \\theta\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5863049c-8e84-4b33-aea7-8865099e96ff",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Proponer un estimador insesgado de $\\theta$. Sugerencia: multiplique por una constante adecuada al estimador propuesto en a.\n",
    "\n",
    "$\\hat{\\theta} = k\\bar{X}(1 - \\bar{X})$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "E(\\hat{\\theta})\n",
    "&= E(k\\bar{X}(1 - \\bar{X})) \\\\\n",
    "&= kE(\\bar{X}(1 - \\bar{X})) \\\\\n",
    "&= k(\\theta - \\tfrac{\\theta}{n}) \\\\\n",
    "&= k\\theta(1 - \\tfrac{1}{n}) \\\\\n",
    "&= k\\theta\\tfrac{n - 1}{n}\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "$\\hat{\\theta}$ es insesgado $\\iff E(\\hat{\\theta}) = \\theta$\n",
    "\n",
    "Busco $k$ tal que $\n",
    "E(\\hat{\\theta})\n",
    "= k\\theta\\tfrac{n - 1}{n}\n",
    "= \\theta\n",
    "\\iff k\\tfrac{n - 1}{n} = 1\n",
    "\\iff k = \\tfrac{n}{n - 1}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4decc8a1-eb4e-409b-85c9-c7c30f1f8381",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "¿Cuál de los dos estimadores tiene menor varianza?\n",
    "\n",
    "Sea $\\hat{\\theta}_{ins} = \\tfrac{n}{n-1}\\hat{\\theta}$ el estimador insesgado.\n",
    "\n",
    "$V(\\hat{\\theta}_{ins}) = V(\\tfrac{n}{n-1}\\hat{\\theta}) = (\\tfrac{n}{n-1})^2 V(\\hat{\\theta})$\n",
    "\n",
    "Observemos que $n > 0 \\implies \\tfrac{n}{n-1} > 1 \\implies (\\tfrac{n}{n-1})^2 > 1 \\implies V(\\hat{\\theta}_{ins}) > V(\\hat{\\theta})$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.1.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}