{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimador de máxima verossimilhança da distribuição Poisson\n",
    "Neste notebook do Jupyter vamos demonstrar a dedução matemática do estimador de máxima verossimilhança do parâmetro da distribuição Poisson. Para ver os resultados dos comandos de cada etapa coloque o mouse ao lado esquerdo da célula que tem os comandos e clique em 'Run this cell'\n",
    "## Preparação\n",
    "Antes de começar importamos a biblioteca de matemática simbólica do Python. Ela permite criar expressões, como funções e suas derivadas, e manipulá-las simbolicamente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A função de densidade probabilística\n",
    "Para uma dada observação $x_i$ a função de massa da distribuição Poisson é"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAxCAYAAABtVXGvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEd0lEQVRoBe2a/1UUMRCAwWcBp3aAHYBWICWAVAB2AM+/4D8fdoBWoNABUIFKB9CBSgf4fSHZl/11HpCF47h5b26yk9lkJjOZ7N7s4tXV1cIsw97e3qH2Qdebdj5rMmbwehOb1jB+rWnbzBuP0ZcY/QX82DR+8bZhz6DbcbDX0H3QlQ1t+i5i31QQ9FlCkXP1y3WreZ6OEXgojtOafg0/gn6GvgQPYvsN7WVwqgDddMYJuJMr1jTeEEl7RG/2wWUc0P4ReBwFN+EfxXYgXKe+nP0QbW3bQh/1DdAZ9gjo+VXoiyjXS5DxuKiFU6/wLTuYw2ga54w08gdkW1sOnvZ8AqUpShf6jF9FSI+tc2PNk/AqiEqdQsMiQZfAMDnUHPAW/E3b7XHvwLx6+RTcoX0CbtHehwZ9a2GftFOQtmHyIfESpU8D/8brDWi+0rU9Rd8P0CR476CeTPoLDIarADyzvjQce8+96AEFtxE0CboQAWhfgN9Bk17Yz7Ftf2U8PBOi/RVPgS5ATg95FP2O/S6YSlfzRv5NyEEcQ0fmYPg719GCR10X7u7uLoFX4HZX//943DcCz5WDrvXJ07eqHLicZGyD++l6KNoZ9nGZ/kTaCv3IH0ui14wSI6S5+uFe+gxNo8N9eBaY1z8p52Ss8s2+hGcYmhmDYtCVhnJFNGFMx9fQfGu84vqYvs4FKzJxHKS155lUb2j4uyjjEaP3bxUBcYw+ouHpYalPZjB+Lewx3PNUb/iw4oOMCcej7j04FHgiPAhUxkfDPRN9UMj3n1nTjB+Oh8Jaekwa5i0YaL7aPMF4JjL8PBP1eG2vxWuVHCLs3VLOXQHzudAueO6Aqr9kIyQ8JjPUfaEJDwHNCeDrdfNA8cdYxvY00Pu+dQkafy9PhJ3Z/lqH2f+t9vzsm9q2cG58e02eBmfu+afh57aVc8+31+RpcBZ5b57tks0YP84fcsYszkx3zRPeTLt3jHFT73ne8M5B3yiLQ+tvrOIz3GFAjF7idtF6YHGYds+nf5ATLboAU208nk9Fi0SfjvHRUg3PS2LFFmCqPR+tNOQHCfvBEh4hO0Jpq6KC1Vrr/iavDVD4gUxvBfhaJPzq+UcX9pagPosaCn4FrflbnXFhWvV2+vzz0gXKwZBPf27m/Du3Bwl7DAh18Ew7Pec/wOnfYY+ulvHwrJ426wOPLux/sgB5klrBqDN4IXyhrW/i6F+An9fsZAneE+4LVwV/hvL8WUNHy13fGrxJL90y+UJOet9/5QZ/pcWbVmQsilQFD3gjNYOmSFDGOuFbeJ1RoXxpKO55DQMtMWuQYJnrkuvcex+5Toa7EMtcmxj9UjIsDO3BYYijTqNFPyvRkNoZDc8+QzmBye8LfBOdHzOERbGTtvf7/Y/5wrxRFIp7Hu0sdJrVNXILpfW8BVA/AfMUMDKq8512Mlg5C5QV0OdCGDE/K2bBxuB7fhJdMTJ52E/E/NqrmTAnGebGMkN4/sZKcINRko7CIUrhnTpNi/FulfTxUtfDT6fyd2X+A2cESc8HN01WAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\lambda^{{x}_{i}} e^{- \\lambda}}{{x}_{i}!}$"
      ],
      "text/plain": [
       "       x[i]  -\\lambda\n",
       "\\lambda    ⋅ℯ        \n",
       "─────────────────────\n",
       "       (x[i])!       "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, L, i, n = symbols('x \\lambda i n')\n",
    "f = L**(Indexed(x,i))*exp(-L)*1/factorial(Indexed(x,i))\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defina a função de log-verossimilhança negativa\n",
    "É a soma dos logarítimos dos valores da função acima sobre todos os valores dos dados ($x_i$). Primeiro calculamos o logarítmo da função de massa:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\lambda + \\log{\\left(\\lambda \\right)} {x}_{i} - \\log{\\left({x}_{i}! \\right)}$"
      ],
      "text/plain": [
       "-\\lambda + log(\\lambda)⋅x[i] - log((x[i])!)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lf = ln(f)\n",
    "lf=expand_log(lf, force=true)\n",
    "lf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e em seguida definimos a sua soma sobre todos os valores de $x_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\lambda \\sum_{i=1}^{n} 1 + \\sum_{i=1}^{n} \\left(\\log{\\left(\\lambda \\right)} {x}_{i} - \\log{\\left({x}_{i}! \\right)}\\right)$"
      ],
      "text/plain": [
       "            n         n                                     \n",
       "           ___       ___                                    \n",
       "           ╲         ╲                                      \n",
       "            ╲         ╲                                     \n",
       "- \\lambda⋅  ╱   1 +   ╱   (log(\\lambda)⋅x[i] - log((x[i])!))\n",
       "           ╱         ╱                                      \n",
       "           ‾‾‾       ‾‾‾                                    \n",
       "          i = 1     i = 1                                   "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LL = Sum(lf, (i,1,n))\n",
    "LL=simplify(LL)\n",
    "LL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O somatório $\\sum_{i=1}^n 1$ nada mais é que $n$. Substituímos isto na função obtida acima temos a função de log-verossimilhança:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\lambda n + \\sum_{i=1}^{n} \\left(\\log{\\left(\\lambda \\right)} {x}_{i} - \\log{\\left({x}_{i}! \\right)}\\right)$"
      ],
      "text/plain": [
       "               n                                     \n",
       "              ___                                    \n",
       "              ╲                                      \n",
       "               ╲                                     \n",
       "-\\lambda⋅n +   ╱   (log(\\lambda)⋅x[i] - log((x[i])!))\n",
       "              ╱                                      \n",
       "              ‾‾‾                                    \n",
       "             i = 1                                   "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g=Sum(1, (i,1,n))\n",
    "LL=LL.subs(g,n)      \n",
    "LL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimativa de máxima verossimilhança (mle)\n",
    "É o valor do parâmetro $\\lambda$ que maximiza a função de log-verossimilhança (ou minimiza a de log-verossimilhança negativa). Sabemos que as funções têm derivadas zero em seus pontos máximos e mínimos.\n",
    "Então o truque para achar a expressão do mle é encontrar a expressão para o valor de $\\lambda$\n",
    "que faz a derivada da função de log-verossimilhança ser igual a zero. São duas etapas:\n",
    "\n",
    "### Calcule a derivada da função de log-verossimlhança em relação ao parâmetro\n",
    "Antes de calcular a derivada temos que substituir o fatorial $x_i!$ pela sua expressão contínua $\\Gamma (x_i+1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAV0AAAA6CAYAAAATDorhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAORUlEQVR4Ae2d/7XUthLHN/dQQHJTQaADEirITQcEKiCvg+Tc//iPQzogVABJBwkVwKODl1cBvNsB7/sRkiPLsiWvZe/u3ZlztLKl8Wj0ndFYln/sF58+fdoZ1SPw9OnTn8T9rdK/ld4pXSlB91T3r8+b9msIGAKGQB6BLyzo5oHJlSqofqnyR0p/K/2u9IvKflO+U/4/Zd8qp87IEDAEDIEsAhfZUiscQ+BSFa+V7iv9HQKuZyYgf/TblhkChoAhkEXgTrbUCrMIhFms8gdieBWYtM8Sw43ym1BmuSFgCBgCOQRspptDpVxGkP0rYvtR28yAdwq8zIKNDAFDwBDIImBBNwvLeKGC6l3Vfqn8fcRFEH6hsodRmW0aAoaAITBAwILuAJJiAUH3j4TrhfYfK10mwThhs11DwBA4dwTs6YVz9wDrvyFgCGyKgM10N4XbGjMEDIFzR8CC7rl7gPXfEDAENkXAgu6mcFtjhoAhcO4IWNA9dw+w/hsChsCmCFjQ3RRua8wQMATOHQELuufuAdZ/Q8AQ2BQBew3Yw63na/lq2Npvk/GBnF83tbA1ZggYAkeFgAXdf8zxTJt8OSwQXwyL3zoL5aO5+PnoDR/FuatEAOcbDbytRjl0rWRB10FhP4bAeSJgL0dEdlfQJOiGV3n5iti9qHrvTcnhG7y/KBGMf9R++kbb3rLtQEPAEDgtBGxNN7KXgiEfrgnfw72r/XjmG3HO25Sc35QI4Hx7d/aHznUswfroaIleS449OiCOUKG18V1b/jFAWtPHEk+u3oLu0LoE3kAPBRqz1CYkWQRcZtDVQVS8P+uYtdeaZ/evgV6c1OibUWMEGthmUqO15U82vm1ljY+WeAb1zZYXZAjWLV+CibbjwLUtTA1a8071PBLFX/GEGXBUvN+mZP2k5P5xYkqCeFjqeKCcpYmdcoIvGBO0/9L+QXBWuz29pEtHqqv2A/ESdPkOcRGLroEj3JD+R2EXoJEuPdt43biXgI6X8Ijiz5Jir+9cqa7ExO98ze8PslT+gOHECjw+XNFyD+cmVV9lRR8t8aT1zYIuyko4BuRva37V9qTx4D9mkv5/Sj9ugkHN1nc/iyv/eizfKOf/2HqkMp60QKfNg+6UXkFJz1PlB74v3ysfOHyQdyq578tB7AJGap/xN+Yz+DI+/Yf4Bn6jMr6Ux1fyBnUqdzQlP/CcQu77weSFf3rhhMMJ6SuVZ31Q5Yy3SR8t8cT1TZcXvNLcJGp2SS5Z1aT2r5RatY3zBSNwiYBTbknMtMfaPOTfAk3p5fCZ6Qf0Mb6q2BLj1m0d0i70Zco2k1dqshlLXwTtKZqSP3Vc8zrpu/dYxz+VuKFNn19VKFfjoyWerr5p0PXKI5yPfHOZszXhNCXHqdIJw4gxPuuzJLBlnx6pvWO87K7Vq8oPfB+R2cRuVca9vUy1thlDYDIw66Cl8sfa3ae82VgvNV7joyWeuL550JVw1osIWLPv0pc6v3W970v8XO1Lla0eHNQGwb00AJrDoXbvTgmdo5d45/gBfX001fY515XsAjZzbDOBJcsPWWokPyv7RAprfLTE4+rXejmCGdrPMhQzXgJwR9pnbYk1lAdKT5RY3CdAf1D6QelP8cSBTkWHI+nCW2RBZwLuG6XBOmtjDcEhvtlRJV56oh+XgP/xB/CY2u8q78ny/QFznIBAy2DjWNYjwX/s+eS5eo36gdqKCf2Q3ZvZSw90CstFwV/Q97ES9FY8R//Ms+/HmnYBi7m24ZgeFbCclH9bbNUDpL+T9dE+ixuzAz+OeJyM5jNd3wCXllAYMJ/3Pv/+IAMRVBlQLGbzWFZ4PZbL+efan5xxiWdr+l4N3vhG70s/BtCaxOJ+CJxV7UgnTmQs+IMfNzJJBFaw7R7N0jZ8BNkn2qbOYa591rkIYFNXKHP1mvIDNdURfc3ZvOuL6t8q4S+s5XGTFv8Z2EF1nOhzssS+PUmXLexCx+baZi4YJfknb6sCIGM+Gh9W4nH1F/ERDbfDDYXeAJYDhjuoNMXAYIDEs1pmvRAD6mhIOhJwn0QKMYtnMK1F9D9gWNsGj71wZ5rZa0wEKAZE0Pda+8xo6VMgzsDuaROVsz1Gc/UKfej5QUY4fL1AKT04YcdBFX0fKoXZML4S12vXEf2D71hoC7vQ17m2mYvPqPxbZKspTAY+mmEu8bj6OwIMMN8okdcSd/7e55i9PBytG+wR7zttM6OiLQZZOhhDYEgDR68pHc8MigCeEgNxp/pULsXvVc6sbi/SsQQ0BnyYvfNozjdKN3sJnD6IflTLlQ7gBp7MBnukOvpN2WOlrM2orKRqvdQmNh7zg7Q57A1/TPhK7Acs6dAXh4vyrC1V7k4esaB0Wzy01cznU/lhX+1sZRearLZN0G9mPiV/TVttPtZHcMn5aMpa4nH1BF2cuMkapWQx8BloXI5DzEQIgC4I+rYodwFT++msygWGiA/eAak+F1R3KmeGw+Nd8ex5cPy+BbSrhO70k4FLAF6lLcmdQ+gDuYD0eXPwSwCAcGLWbbnKCPzU5WaN8M8myZ30g4zAwYCWjPQEwY22Z5ljZxf5fjfx+ULjR2WXgq57V69sq4OM9QwYAx/dg8fJuJM5cK8iAc/AJeAyC3aDWTlrhAyWFLixWRdB0/Hq2DgoqPhoiIVy1ma4RF8r4HIZQlCvpTAjnDom8LyTUGbsPIlBGYGB5Ydw2a7dUSrqJTlz/CA0hN7IzpJkcqKDp7tppjLXV+U3HOR5aJs3+LKzYPg2poC503Wk7cCzxC6ILtpmpP3a4ir53g6naKsSDvRp1Ef9wSUeV39RaqmmXkDj7FyuMRN8Hx3DrIrgma6xMYh6s9yI57U//jqSc0yb9OVGieC7FoVgWCXfY57VSXVgDXFChNjniQZOju5GmvKagMuxk3pJzlw/QCZ0qRSCz05y8Blm40F3TsQ32u94tH+tffq8U44zc4OTkyA3Ztk/OEkPxsIWdqGvk7ZpAEZWPlgrnbytKvDp+egIf4nH1S8OugKcgcFdc+6Gp4GUfYzVzXTFw8yKQfFKKSbK3ZqdeLhsT+tj3oNs+75yGc4rgXEAaK0Pg/XBTKEs6fDw+v3kOPTlSYZgG4KAu7GmMpZiSLVBalQvyZjlB4mObr02KkMW6aPX7WNUt/NtvY3KcGa+G8AJMb1JGLEdZHMLu9CxUdv4Xgcbh3wuGGPyT91WX3sg8KEpSn00x1vicfV3ckfOLOPGBTPc7tIvOZ56ZlYM7nC2ZGBgxJiYbXFpSMBlsKX1Me/m29KHYMajVswQ19aNE06YmXZ99TpwBfAdhdqHh9mqw1M5RmWfwApxImM/BNwd20rUcaLsSGXY5oXyqSWTrF5eyFw/6NrWBgO3OzFrG33xB8p30gn/YjbFlRN64x+dv2nbnQCVIwOeTUnt4huHtAv9zdom0g1fgK5UFnzrmbZrfTkrX/JOylYOAf1EGDgfUxExCj/Cz3JXfqmPBlFxXuJx9U0/eBO3fohtgcVMp/mNNMlldvBfJZx0Kig167baYd24eYCXXE4cBFcXtJTTN87yBI7nvm60j631kjyCAY5+T/ne5PvBR3a+UsIHaoPJ3m22PFD6LrILukjGKj4T+tlKvuTgc4tsJRmrjPXQ1zhXW0UfLfHE9Rex8FuwzQyP1JpYr34t4EaDUesGJY8AGM/+Fjch/d1ZXXk8S3Rrpb6MNktr1a31YoaMzKVE39zylPKmuC1VrHR8I7vQTGvbpKq3kt/CVmuN9bTP7Nf4aImnq79VQVfOy6Vz7tIgB2RVmeQxA+FydtOB7PvBjI2zbCt6J0HIZFabo+LleUu9fN/Qp4XNuMxlmYW371oE8Rw+a5UttguKeRxb+0zX54byF9tKujQf611How21w/ib9NEST1p/J5JvmwkCAov1QUBnrbQZSS532LvZ5oRgHn1i/a00+5wQ8U+V2mRWS1+48/9Y+Qdf+7VyLvm4GVpzWd5KL/BtcjKjb5J1LI+KSZV6QnelFnah0Va2GevAYvknZqsaHy3x9Opv1ZrumJfsUy7H4IYeMya+KO9u1OwjJz3Gy+3dCEp54n3xE/QJ0lsubcQqZLeX6qXjmZHyll8zbLOKnmHhUtuUIFtbfqn9reprfLTEk6u3oJuxoIBizYllhdl/w54R1xVJLov/L5Vzs8fIEDAEzhABW15IjK6AyMySgNvsyQHJ5NKdWTOz5xbrlxJjZAgYAqeIgAXdyGo+OPIcKM+21qy5Rkf3N33w5oYVa6fMcAOxvmNkCBgCZ4qALS9EhlegJOASKGtuJkVHuk1ms9ClUth2BdEPd9cXPY8aybJNQ8AQOEEEbKbrjaZgyAw0PEoV8tYmPbVHmVr33+QZAmePgM10z94FDABDwBDYEoGLLRuztgwBQ8AQOHcELOieuwdY/w0BQ2BTBCzobgq3NWYIGALnjoAF3REP0I01Ps58d6Taig0BQ8AQ2AsBe3phHLZrVfFtgkWv3xK8JcP9ZZG2m37DYVx1qzEEDIFjRcCC7ohlFCD5FNsikgwePbvyQsae3V3Uhh1sCBgCp4WABd0V7aWgy0sWfOM1fiNtxRZNtCFgCBw7AhZ0EwspQDIzZYbKXwed5KcCky7ZriFgCBwRAhZ0I2MoyLIE4P5VVtuf2Fe6gUU5b6zV3Fjjv5bsozaAZmQIGAIDBCzo9iG51G72X2UVSJt8bLvfnO0ZAobAuSFgQTeyuALrQf9VNlLFNg0BQ+CWImBBNzGsAi9LDFdKfE+XpQb3xTHltryQYGW7hoAhMB8BC7pDzAi47l9lFWhZUnDLCn57yG0lhoAhYAjMQMC+MpaApeDKTPel0lulRf/hJVnceCNoE8h5IoIXLT6ofNELF5JhZAgYAieKwP8BWosul4Z3XLgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle - \\lambda n + \\sum_{i=1}^{n} \\left(\\log{\\left(\\lambda \\right)} {x}_{i} - \\log{\\left(\\Gamma\\left({x}_{i} + 1\\right) \\right)}\\right)$"
      ],
      "text/plain": [
       "               n                                         \n",
       "              ___                                        \n",
       "              ╲                                          \n",
       "               ╲                                         \n",
       "-\\lambda⋅n +   ╱   (log(\\lambda)⋅x[i] - log(Γ(x[i] + 1)))\n",
       "              ╱                                          \n",
       "              ‾‾‾                                        \n",
       "             i = 1                                       "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LL=simplify(LL)\n",
    "LL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "E agora usamos o comando 'diff' para obter a derivada da log-verossimilhança em função do parâmetro $\\lambda$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - n + \\frac{\\sum_{i=1}^{n} {x}_{i}}{\\lambda}$"
      ],
      "text/plain": [
       "       n       \n",
       "      ___      \n",
       "      ╲        \n",
       "       ╲       \n",
       "       ╱   x[i]\n",
       "      ╱        \n",
       "      ‾‾‾      \n",
       "     i = 1     \n",
       "-n + ──────────\n",
       "      \\lambda  "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dLL = diff(LL,L)\n",
    "dLL=simplify(expand(dLL))\n",
    "dLL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolvemos a derivada para o parâmetro\n",
    "Agora igualamos a derivada da função de log-verossimilhança negativa a zero e resolvemos esta equação para o parãmetro $\\lambda$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\sum_{i=1}^{n} {x}_{i}}{n}\\right]$"
      ],
      "text/plain": [
       "⎡  n       ⎤\n",
       "⎢ ___      ⎥\n",
       "⎢ ╲        ⎥\n",
       "⎢  ╲       ⎥\n",
       "⎢  ╱   x[i]⎥\n",
       "⎢ ╱        ⎥\n",
       "⎢ ‾‾‾      ⎥\n",
       "⎢i = 1     ⎥\n",
       "⎢──────────⎥\n",
       "⎣    n     ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(dLL,L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Que nada mais é que a média amostral!"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}