{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Distribuciones de probabilidad con Python" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*Esta notebook fue creada originalmente como un blog post por [Raúl E. López Briega](http://relopezbriega.com.ar/) en [Matemáticas, análisis de datos y python](http://relopezbriega.github.io). El contenido esta bajo la licencia BSD.*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\"Distribuciones" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Introducción\n", "\n", "Las [variables aleatorias](https://es.wikipedia.org/wiki/Variable_aleatoria) han llegado a desempeñar un papel importante en casi todos los campos de estudio: en la [Física](https://es.wikipedia.org/wiki/F%C3%ADsica), la [Química](https://es.wikipedia.org/wiki/Qu%C3%ADmica) y la [Ingeniería](https://es.wikipedia.org/wiki/Ingenier%C3%ADa); y especialmente en las ciencias biológicas y sociales. Estas [variables aleatorias](https://es.wikipedia.org/wiki/Variable_aleatoria) son medidas y analizadas en términos\n", "de sus propiedades [estadísticas](https://es.wikipedia.org/wiki/Estad%C3%ADstica) y [probabilísticas](https://es.wikipedia.org/wiki/Probabilidad), de las cuales una característica subyacente es su [función de distribución](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_distribuci%C3%B3n). A pesar de que el número potencial de [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) puede ser muy grande, en la práctica, un número relativamente pequeño se utilizan; ya sea porque tienen características matemáticas que las hace fáciles de usar o porque se asemejan bastante bien a una porción de la realidad, o por ambas razones combinadas.\n", "\n", "## ¿Por qué es importante conocer las distribuciones?\n", "\n", "Muchos resultados en las ciencias se basan en conclusiones que se extraen sobre una población general a partir del estudio de una *[muestra](https://es.wikipedia.org/wiki/Muestra_estad%C3%ADstica)* de esta población. Este proceso se conoce como ***[inferencia estadística](https://es.wikipedia.org/wiki/Estad%C3%ADstica_inferencial)***; y este tipo de *inferencia* con frecuencia se basa en hacer suposiciones acerca de la forma en que los datos se distribuyen, o requiere realizar alguna transformación de los datos para que se ajusten mejor a alguna de las [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) conocidas y estudiadas en profundidad.\n", "\n", "Las [distribuciones de probabilidad](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) teóricas son útiles en la [inferencia estadística](https://es.wikipedia.org/wiki/Estad%C3%ADstica_inferencial) porque sus propiedades y características son conocidas. Si la [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) real de un [conjunto de datos](https://es.wikipedia.org/wiki/Conjunto_de_datos) dado es razonablemente cercana a la de una [distribución de probabilidad](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) teórica, muchos de los cálculos se pueden realizar en los datos reales utilizando hipótesis extraídas de la [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) teórica.\n", " \n", "## Graficando distribuciones\n", "\n", "### Histogramas\n", "\n", "Una de las mejores maneras de describir una variable es representar los valores que aparecen en el [conjunto de datos](https://es.wikipedia.org/wiki/Conjunto_de_datos) y el número de veces que aparece cada valor. La representación más común de una [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) es un [histograma](https://es.wikipedia.org/wiki/Histograma), que es un gráfico que muestra la frecuencia de cada valor.\n", "\n", "En [Python](http://python.org/), podemos graficar fácilmente un histograma con la ayuda de la función `hist` de [matplotlib](http://matplotlib.org/api/pyplot_api.html), simplemente debemos pasarle los datos y la cantidad de *contenedores* en los que queremos dividirlos. Por ejemplo, podríamos graficar el [histograma](https://es.wikipedia.org/wiki/Histograma) de una [distribución normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal) del siguiente modo." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "hide_input": false }, "outputs": [], "source": [ "# \n", "# importando modulos necesarios\n", "%matplotlib inline\n", "\n", "import matplotlib.pyplot as plt\n", "import numpy as np \n", "from scipy import stats \n", "import seaborn as sns \n", "\n", "np.random.seed(2016) # replicar random\n", "\n", "# parametros esteticos de seaborn\n", "sns.set_palette(\"deep\", desat=.6)\n", "sns.set_context(rc={\"figure.figsize\": (8, 4)})" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Lli2y2WxKTU3VQw895MdqzVefKdENw9ADDzyg3r17a+jQoX6q1Ds89Z+Xl+ee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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando histograma\n", "mu, sigma = 0, 0.2 # media y desvio estandar\n", "datos = np.random.normal(mu, sigma, 1000) #creando muestra de datos\n", "\n", "# histograma de distribución normal.\n", "cuenta, cajas, ignorar = plt.hist(datos, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Función de Masa de Probabilidad\n", "\n", "Otra forma de representar a las [distribuciones discretas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta) es utilizando su [Función de Masa de Probabilidad](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad) o [FMP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad), la cual relaciona cada valor con su *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* en lugar de su *frecuencia* como vimos anteriormente. Esta función es *normalizada* de forma tal que el valor total de *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* sea 1. La ventaja que nos ofrece utilizar la [FMP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad) es que podemos comparar dos [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) sin necesidad de ser confundidos por las diferencias en el tamaño de las *[muestras](https://es.wikipedia.org/wiki/Muestra_estad%C3%ADstica)*. También debemos tener en cuenta que [FMP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad) funciona bien si el número de valores es pequeño; pero a medida que el número de valores aumenta, la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* asociada a cada valor se hace cada vez más pequeña y el efecto del *ruido aleatorio* aumenta. \n", "Veamos un ejemplo con [Python](http://python.org/)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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v3op0Wg2ps+IZNKC32lG6nJkmajU7+deaHZyukL8x0XmkcnRzpZXlHDlTQIhf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\nUlJS2L59OwCXLl1i7NixpKamkpqayrJ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando FMP\n", "n, p = 30, 0.4 # parametros de forma de la distribución binomial\n", "n_1, p_1 = 20, 0.3 # parametros de forma de la distribución binomial\n", "x = np.arange(stats.binom.ppf(0.01, n, p),\n", " stats.binom.ppf(0.99, n, p))\n", "x_1 = np.arange(stats.binom.ppf(0.01, n_1, p_1),\n", " stats.binom.ppf(0.99, n_1, p_1))\n", "fmp = stats.binom.pmf(x, n, p) # Función de Masa de Probabilidad\n", "fmp_1 = stats.binom.pmf(x_1, n_1, p_1) # Función de Masa de Probabilidad\n", "plt.plot(x, fmp, '--')\n", "plt.plot(x_1, fmp_1)\n", "plt.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "plt.vlines(x_1, 0, fmp_1, colors='g', lw=5, alpha=0.5)\n", "plt.title('Función de Masa de Probabilidad')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Función de Distribución Acumulada\n", "\n", "Si queremos evitar los problemas que se generan con [FMP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad) cuando el número de valores es muy grande, podemos recurrir a utilizar la [Función de Distribución Acumulada](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_distribuci%C3%B3n) o [FDA](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_distribuci%C3%B3n), para representar a nuestras [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad), tanto [discretas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta) como [continuas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad_continua). Esta función relaciona los valores con su correspondiente [percentil](https://es.wikipedia.org/wiki/Percentil); es decir que va a describir la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de que una [variable aleatoria](https://es.wikipedia.org/wiki/Variable_aleatoria) X sujeta a cierta ley de [distribución de probabilidad](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) se sitúe en la zona de valores menores o iguales a x." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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UdiHuQb2+nkNZh3FxdCFRho69rS17TnMs7TLurk48MWMUYcHdbB1JiC6l1cK+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\n6mrJyblMjx63vv/ufhfQ8eNHKS4uYsW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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Función de Distribución Acumulada con Python\n", "x_1 = np.linspace(stats.norm(10, 1.2).ppf(0.01),\n", " stats.norm(10, 1.2).ppf(0.99), 100)\n", "fda_binom = stats.binom.cdf(x, n, p) # Función de Distribución Acumulada\n", "fda_normal = stats.norm(10, 1.2).cdf(x_1) # Función de Distribución Acumulada\n", "plt.plot(x, fda_binom, '--', label='FDA binomial')\n", "plt.plot(x_1, fda_normal, label='FDA nomal')\n", "plt.title('Función de Distribución Acumulada')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.legend(loc=4)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Función de Densidad de Probabilidad \n", "\n", "Por último, el equivalente a la [FMP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_probabilidad) para [distribuciones continuas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad_continua) es la [Función de Densidad de Probabilidad](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_densidad_de_probabilidad) o [FDP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_densidad_de_probabilidad). Esta función es la [derivada](https://es.wikipedia.org/wiki/Derivada) de la [Función de Distribución Acumulada](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_distribuci%C3%B3n).\n", "Por ejemplo, para la [distribución normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal) que graficamos anteriormente, su [FDP](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_densidad_de_probabilidad) es la siguiente. La típica forma de campana que caracteriza a esta [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad)." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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TNntfX1/i4+OZMGEC6enpeHreeR/ySy+9RFBQEAsWLGj429KlS3F2dmbBggVk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\ntSO1eNLsW5Gq6lq+/DaBS9dKCejfnYi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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Función de Densidad de Probibilidad con Python\n", "FDP_normal = stats.norm(10, 1.2).pdf(x_1) # FDP\n", "plt.plot(x_1, FDP_normal, label='FDP nomal')\n", "plt.title('Función de Densidad de Probabilidad')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Distribuciones\n", "\n", "Ahora que ya conocemos como podemos hacer para representar a las [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad); pasemos a analizar cada una de ellas en más detalle para conocer su forma, sus principales aplicaciones y sus propiedades. Comencemos por las [distribuciones discretas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta).\n", "\n", "## Distribuciones Discretas\n", "\n", "Las [distribuciones discretas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta) son aquellas en las que la variable puede tomar solo algunos valores determinados. Los principales exponentes de este grupo son las siguientes: \n", "\n", "### Distribución Poisson\n", "\n", "La [Distribución Poisson](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Poisson) esta dada por la formula:\n", "\n", "$$p(r; \\mu) = \\frac{\\mu^r e^{-\\mu}}{r!}$$\n", "\n", "En dónde $r$ es un [entero](https://es.wikipedia.org/wiki/N%C3%BAmero_entero) ($r \\ge 0$) y $\\mu$ es un [número real](https://es.wikipedia.org/wiki/N%C3%BAmero_real) positivo. La [Distribución Poisson](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Poisson) describe la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de encontrar exactamente $r$ eventos en un lapso de tiempo si los acontecimientos se producen de forma independiente a una velocidad constante $\\mu$. Es una de las [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) más utilizadas en [estadística](http://relopezbriega.github.io/tag/estadistica.html) con varias aplicaciones; como por ejemplo describir el número de fallos en un lote de materiales o la cantidad de llegadas por hora a un centro de servicios. \n", "\n", "En [Python](http://python.org/) la podemos generar fácilmente con la ayuda de [scipy.stats](http://docs.scipy.org/doc/scipy/reference/stats.html), paquete que utilizaremos para representar a todas las restantes [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) a lo largo de todo el artículo." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Poisson\n", "mu = 3.6 # parametro de forma \n", "poisson = stats.poisson(mu) # Distribución\n", "x = np.arange(poisson.ppf(0.01),\n", " poisson.ppf(0.99))\n", "fmp = poisson.pmf(x) # Función de Masa de Probabilidad\n", "plt.plot(x, fmp, '--')\n", "plt.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "plt.title('Distribución Poisson')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = poisson.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Poisson')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Distribución Binomial\n", "\n", "La [Distribución Binomial](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_binomial) esta dada por la formula:\n", "\n", "$$p(r; N, p) = \\left(\\begin{array}{c} N \\\\ r \\end{array}\\right) p^r(1 - p)^{N - r}\n", "$$\n", "\n", "En dónde $r$ con la condición $0 \\le r \\le N$ y el parámetro $N$ ($N > 0$) son [enteros](https://es.wikipedia.org/wiki/N%C3%BAmero_entero); y el parámetro $p$ ($0 \\le p \\le 1$) es un [número real](https://es.wikipedia.org/wiki/N%C3%BAmero_real). La [Distribución Binomial](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_binomial) describe la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de exactamente $r$ éxitos en $N$ pruebas si la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de éxito en una sola prueba es $p$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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kEgk+/fRTDBjAaUqp7e3YexxnLl7DAK9ueDT4AbHjEHUYggAUFd/E2YJr2Lk3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\nSfHw8MCyZcv010+cOKGf3yE0NBT79+8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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Binomial\n", "N, p = 30, 0.4 # parametros de forma \n", "binomial = stats.binom(N, p) # Distribución\n", "x = np.arange(binomial.ppf(0.01),\n", " binomial.ppf(0.99))\n", "fmp = binomial.pmf(x) # Función de Masa de Probabilidad\n", "plt.plot(x, fmp, '--')\n", "plt.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "plt.title('Distribución Binomial')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = binomial.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Binomial')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Distribución Geométrica\n", "\n", "La [Distribución Geométrica](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_geom%C3%A9trica) esta dada por la formula:\n", "\n", "$$p(r; p) = p(1- p)^{r-1}\n", "$$\n", "\n", "En dónde $r \\ge 1$ y el parámetro $p$ ($0 \\le p \\le 1$) es un [número real](https://es.wikipedia.org/wiki/N%C3%BAmero_real). La [Distribución Geométrica](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_geom%C3%A9trica) expresa la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de tener que esperar exactamente $r$ pruebas hasta encontrar el primer éxito si la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de éxito en una sola prueba es $p$. Por ejemplo, en un proceso de selección, podría definir el número de entrevistas que deberíamos realizar antes de encontrar al primer candidato aceptable." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Geométrica\n", "p = 0.3 # parametro de forma \n", "geometrica = stats.geom(p) # Distribución\n", "x = np.arange(geometrica.ppf(0.01),\n", " geometrica.ppf(0.99))\n", "fmp = geometrica.pmf(x) # Función de Masa de Probabilidad\n", "plt.plot(x, fmp, '--')\n", "plt.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "plt.title('Distribución Geométrica')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = geometrica.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Geométrica')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución Hipergeométrica\n", "\n", "La [Distribución Hipergeométrica](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_hipergeom%C3%A9trica) esta dada por la formula:\n", "\n", "$$p(r; n, N, M) = \\frac{\\left(\\begin{array}{c} M \\\\ r \\end{array}\\right)\\left(\\begin{array}{c} N - M\\\\ n -r \\end{array}\\right)}{\\left(\\begin{array}{c} N \\\\ n \\end{array}\\right)}\n", "$$\n", "\n", "En dónde el valor de $r$ esta limitado por $\\max(0, n - N + M)$ y $\\min(n, M)$ inclusive; y los parámetros $n$ ($1 \\le n \\le N$), $N$ ($N \\ge 1$) y $M$ ($M \\ge 1$) son todos [números enteros](https://es.wikipedia.org/wiki/N%C3%BAmero_entero). La [Distribución Hipergeométrica](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_hipergeom%C3%A9trica) describe experimentos en donde se seleccionan los elementos al azar *sin reemplazo* (se evita seleccionar el mismo elemento más de una vez). Más precisamente, supongamos que tenemos $N$ elementos de los cuales $M$ tienen un cierto atributo (y $N - M$ no tiene). Si escogemos $n$ elementos al azar *sin reemplazo*, $p(r)$ es la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de que exactamente $r$ de los elementos seleccionados provienen del grupo con el atributo. " ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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dmVr2Ijei2OEgcXMa3l5Wet2qVn1V1bheHRrXq2N0DJFqrUzFfsyYMXTr1o1N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\nJCVZgDql+kWo2Et1olnv5LJ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Hipergeométrica\n", "M, n, N = 30, 10, 12 # parametros de forma \n", "hipergeometrica = stats.hypergeom(M, n, N) # Distribución\n", "x = np.arange(0, n+1)\n", "fmp = hipergeometrica.pmf(x) # Función de Masa de Probabilidad\n", "plt.plot(x, fmp, '--')\n", "plt.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "plt.title('Distribución Hipergeométrica')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = hipergeometrica.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Hipergeométrica')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución de Bernoulli\n", "\n", "La [Distribución de Bernoulli](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Bernoulli) esta dada por la formula:\n", "\n", "$$p(r;p) = \\left\\{\n", "\t\\begin{array}{ll}\n", " 1 - p = q & \\mbox{si } r = 0 \\ \\mbox{(fracaso)}\\\\\n", " p & \\mbox{si } r = 1 \\ \\mbox{(éxito)}\n", "\t\\end{array}\n", "\\right.$$\n", "\n", "En dónde el parámetro $p$ es la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de éxito en un solo ensayo, la *[probabilidad](https://es.wikipedia.org/wiki/Probabilidad)* de fracaso por lo tanto va a ser $1 - p$ (muchas veces expresada como $q$). Tanto $p$ como $q$ van a estar limitados al intervalo de cero a uno. La [Distribución de Bernoulli](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Bernoulli) describe un experimento probabilístico en donde el ensayo tiene dos posibles resultados, éxito o fracaso. Desde esta [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) se pueden deducir varias [Funciones de Densidad de Probabilidad](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_densidad_de_probabilidad) de otras [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) que se basen en una serie de ensayos independientes." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Bernoulli\n", "p = 0.5 # parametro de forma \n", "bernoulli = stats.bernoulli(p)\n", "x = np.arange(-1, 3)\n", "fmp = bernoulli.pmf(x) # Función de Masa de Probabilidad\n", "fig, ax = plt.subplots()\n", "ax.plot(x, fmp, 'bo')\n", "ax.vlines(x, 0, fmp, colors='b', lw=5, alpha=0.5)\n", "ax.set_yticks([0., 0.2, 0.4, 0.6])\n", "plt.title('Distribución Bernoulli')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = bernoulli.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Bernoulli')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Distribuciones continuas\n", "\n", "Ahora que ya conocemos las principales [distribuciones discretas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta), podemos pasar a describir a las [distribuciones continuas](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad_continua); en ellas a diferencia de lo que veíamos antes, la variable puede tomar cualquier valor dentro de un intervalo específico. Dentro de este grupo vamos a encontrar a las siguientes: \n", "\n", "### Distribución de Normal\n", "\n", "La [Distribución Normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal), o también llamada [Distribución de Gauss](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal), es aplicable a un amplio rango de problemas, lo que la convierte en la [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) más utilizada en [estadística](http://relopezbriega.github.io/tag/estadistica.html); esta dada por la formula:\n", "\n", "$$p(x;\\mu, \\sigma^2) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{\\frac{-1}{2}\\left(\\frac{x - \\mu}{\\sigma} \\right)^2}\n", "$$\n", "\n", "En dónde $\\mu$ es el parámetro de ubicación, y va a ser igual a la [media aritmética](https://es.wikipedia.org/wiki/Media_aritm%C3%A9tica) y $\\sigma^2$ es el [desvío estándar](https://es.wikipedia.org/wiki/Desviaci%C3%B3n_t%C3%ADpica). Algunos ejemplos de variables asociadas a fenómenos naturales que siguen el modelo de la [Distribución Normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_normal) son:\n", "* características morfológicas de individuos, como la estatura;\n", "* características sociológicas, como el consumo de cierto producto por un mismo grupo de individuos;\n", "* características psicológicas, como el cociente intelectual;\n", "* nivel de ruido en telecomunicaciones;\n", "* errores cometidos al medir ciertas magnitudes;\n", "* etc." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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aB0/l46XYcKEjkQHQqNxffvllBAcHIyMjAyKRCH//+9/h4eGh7WxEz+TQqXyo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\n6EjUx1ju9MzyS67jbO4lDB0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Normal\n", "mu, sigma = 0, 0.2 # media y desvio estandar\n", "normal = stats.norm(mu, sigma)\n", "x = np.linspace(normal.ppf(0.01),\n", " normal.ppf(0.99), 100)\n", "fp = normal.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Normal')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = normal.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Normal')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución Uniforme\n", "\n", "La [Distribución Uniforme](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_uniforme_discreta) es un caso muy simple expresada por la función:\n", "\n", "$$f(x; a, b) = \\frac{1}{b -a} \\ \\mbox{para} \\ a \\le x \\le b\n", "$$\n", "\n", "Su [función de distribución](https://es.wikipedia.org/wiki/Funci%C3%B3n_de_distribuci%C3%B3n) esta entonces dada por:\n", "\n", "$$\n", "p(x;a, b) = \\left\\{\n", "\t\\begin{array}{ll}\n", " 0 & \\mbox{si } x \\le a \\\\\n", " \\frac{x-a}{b-a} & \\mbox{si } a \\le x \\le b \\\\\n", " 1 & \\mbox{si } b \\le x\n", "\t\\end{array}\n", "\\right.\n", "$$\n", "\n", "Todos los valore tienen prácticamente la misma probabilidad." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Uniforme\n", "uniforme = stats.uniform()\n", "x = np.linspace(uniforme.ppf(0.01),\n", " uniforme.ppf(0.99), 100)\n", "fp = uniforme.pdf(x) # Función de Probabilidad\n", "fig, ax = plt.subplots()\n", "ax.plot(x, fp, '--')\n", "ax.vlines(x, 0, fp, colors='b', lw=5, alpha=0.5)\n", "ax.set_yticks([0., 0.2, 0.4, 0.6, 0.8, 1., 1.2])\n", "plt.title('Distribución Uniforme')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = uniforme.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Uniforme')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Distribución de Log-normal\n", "\n", "La [Distribución Log-normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_log-normal) esta dada por la formula:\n", "\n", "$$p(x;\\mu, \\sigma) = \\frac{1}{ x \\sigma \\sqrt{2 \\pi}} e^{\\frac{-1}{2}\\left(\\frac{\\ln x - \\mu}{\\sigma} \\right)^2}\n", "$$\n", "\n", "En dónde la variable $x > 0$ y los parámetros $\\mu$ y $\\sigma > 0$ son todos [números reales](https://es.wikipedia.org/wiki/N%C3%BAmero_real). La [Distribución Log-normal](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_log-normal) es aplicable a [variables aleatorias](https://es.wikipedia.org/wiki/Variable_aleatoria) que están limitadas por cero, pero tienen pocos valores grandes. Es una [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) con [asimetría positiva](https://es.wikipedia.org/wiki/Asimetr%C3%ADa_estad%C3%ADstica). Algunos de los ejemplos en que la solemos encontrar son:\n", "* El peso de los adultos.\n", "* La concentración de los minerales en depósitos.\n", "* Duración de licencia por enfermedad.\n", "* Distribución de riqueza\n", "* Tiempos muertos de maquinarias." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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wFEXhs69S5NI4IYRoQosG1D366KPMnj0bi8XCypUrOXr0KOPHj3d2Npdmq62j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\n0bn961XX1FJSVsmVskpKyqq4Ul7JlbI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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Log-Normal\n", "sigma = 0.6 # parametro\n", "lognormal = stats.lognorm(sigma)\n", "x = np.linspace(lognormal.ppf(0.01),\n", " lognormal.ppf(0.99), 100)\n", "fp = lognormal.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Log-normal')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = lognormal.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Log-normal')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Distribución de Exponencial\n", "\n", "La [Distribución Exponencial](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_exponencial) esta dada por la formula:\n", "\n", "$$p(x;\\alpha) = \\frac{1}{ \\alpha} e^{\\frac{-x}{\\alpha}}\n", "$$\n", "\n", "En dónde tanto la variable $x$ como el parámetro $\\alpha$ son [números reales](https://es.wikipedia.org/wiki/N%C3%BAmero_real) positivos. La [Distribución Exponencial](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_exponencial) tiene bastantes aplicaciones, tales como la desintegración de un átomo radioactivo o el tiempo entre eventos en un proceso de [Poisson](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Poisson) donde los acontecimientos suceden a una velocidad constante." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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QYuT6RYQGM238IP756TfMW/4dNquFrh2j3V2WiIgYqNbD7ImJia6fkpISkpOT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\nbauw67oRjIh4Fh1mlxp02Kx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=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Exponencial\n", "exponencial = stats.expon()\n", "x = np.linspace(exponencial.ppf(0.01),\n", " exponencial.ppf(0.99), 100)\n", "fp = exponencial.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Exponencial')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = exponencial.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Exponencial')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Distribución Gamma\n", "\n", "La [Distribución Gamma](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_gamma) esta dada por la formula:\n", "\n", "$$p(x;a, b) = \\frac{a(a x)^{b -1} e^{-ax}}{\\Gamma(b)}\n", "$$\n", "\n", "En dónde los parámetros $a$ y $b$ y la variable $x$ son [números reales](https://es.wikipedia.org/wiki/N%C3%BAmero_real) positivos y $\\Gamma(b)$ es la [función gamma](https://es.wikipedia.org/wiki/Funci%C3%B3n_gamma). La [Distribución Gamma](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_gamma) comienza en el *origen* de coordenadas y tiene una forma bastante flexible. Otras [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) son casos especiales de ella." ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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EpVLRu3dvtFqzTmNvcYxGI5k5V7C30xHQxV3pOMIC9Q7qwqQHwtnxTTL/2ZbA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\nvZ4rReUE+3uj1WiUjiOExdB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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Gamma\n", "a = 2.6 # parametro de forma.\n", "gamma = stats.gamma(a)\n", "x = np.linspace(gamma.ppf(0.01),\n", " gamma.ppf(0.99), 100)\n", "fp = gamma.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Gamma')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = gamma.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Gamma')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución Beta\n", "\n", "La [Distribución Beta](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_beta) esta dada por la formula:\n", "\n", "$$p(x;p, q) = \\frac{1}{B(p, q)} x^{p-1}(1 - x)^{q-1}\n", "$$\n", "\n", "En dónde los parámetros $p$ y $q$ son [números reales](https://es.wikipedia.org/wiki/N%C3%BAmero_real) positivos, la variable $x$ satisface la condición $0 \\le x \\le 1$ y $B(p, q)$ es la [función beta](https://es.wikipedia.org/wiki/Funci%C3%B3n_beta). Las aplicaciones de la [Distribución Beta](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_beta) incluyen el modelado de [variables aleatorias](https://es.wikipedia.org/wiki/Variable_aleatoria) que tienen un rango finito de $a$ hasta $b$. Un\n", "ejemplo de ello es la distribución de los tiempos de actividad en las redes de proyectos. La [Distribución Beta](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_beta) se utiliza también con frecuencia como una [probabilidad a priori](https://es.wikipedia.org/wiki/Probabilidad_a_priori) para proporciones [binomiales]((https://es.wikipedia.org/wiki/Distribuci%C3%B3n_binomial) en el [análisis bayesiano](https://es.wikipedia.org/wiki/Inferencia_bayesiana)." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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5AHdGNfVJPCzoREREt+DcpULs3n8ch0+chyTLiAg1YkD8/Xigw90eWTDGWSzo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\nGenp6VCpVHjyySf9sqA3lMO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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Beta\n", "a, b = 2.3, 0.6 # parametros de forma.\n", "beta = stats.beta(a, b)\n", "x = np.linspace(beta.ppf(0.01),\n", " beta.ppf(0.99), 100)\n", "fp = beta.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Beta')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = beta.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Beta')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución Chi cuadrado\n", "\n", "La [Distribución Chi cuadrado](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_%CF%87%C2%B2) esta dada por la función:\n", "\n", "$$p(x; n) = \\frac{\\left(\\frac{x}{2}\\right)^{\\frac{n}{2}-1} e^{\\frac{-x}{2}}}{2\\Gamma \\left(\\frac{n}{2}\\right)}\n", "$$\n", "\n", "En dónde la variable $x \\ge 0$ y el parámetro $n$, el número de grados de libertad, es un [número entero](https://es.wikipedia.org/wiki/N%C3%BAmero_entero) positivo. Una importante aplicación de la [Distribución Chi cuadrado](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_%CF%87%C2%B2) es que cuando un [conjunto de datos](https://es.wikipedia.org/wiki/Conjunto_de_datos) es representado por un modelo teórico, esta [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) puede ser utilizada para controlar cuan bien se ajustan los valores predichos por el modelo, y los datos realmente observados." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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fP8+vf/1rmpqa8PLy4qWXXsLNze02yxTWcvWofmCgHKsX1jNx1CCKSs+Qfeg4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\n7eBsxSXe+WAHjSYz86ePpn8fX7UjCSG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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Chi cuadrado\n", "df = 34 # parametro de forma.\n", "chi2 = stats.chi2(df)\n", "x = np.linspace(chi2.ppf(0.01),\n", " chi2.ppf(0.99), 100)\n", "fp = chi2.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución Chi cuadrado')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = chi2.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma Chi cuadrado')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución T de Student\n", "\n", "La [Distribución t de Student](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_t_de_Student) esta dada por la función:\n", "\n", "$$p(t; n) = \\frac{\\Gamma(\\frac{n+1}{2})}{\\sqrt{n\\pi}\\Gamma(\\frac{n}{2})} \\left( 1 + \\frac{t^2}{2} \\right)^{-\\frac{n+1}{2}}\n", "$$\n", "\n", "En dónde la variable $t$ es un [número real](https://es.wikipedia.org/wiki/N%C3%BAmero_real) y el parámetro $n$ es un [número entero](https://es.wikipedia.org/wiki/N%C3%BAmero_entero) positivo. La [Distribución t de Student](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_t_de_Student) es utilizada para probar si la diferencia entre las *medias* de dos muestras de observaciones es estadísticamente significativa. Por ejemplo, las alturas de una muestra aleatoria de los jugadores de baloncesto podría compararse con las alturas de una muestra aleatoria de jugadores de fútbol; esta [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) nos podría ayudar a determinar si un grupo es significativamente más alto que el otro." ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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8+eWXAMybN49HH30Ue3t7goOD+fDDDy0ZTQizGAwGtqbloYLbP+AyqI34AX29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\nwS9ZsuSuf3/22WctEkYIcxi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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando t de Student\n", "df = 50 # parametro de forma.\n", "t = stats.t(df)\n", "x = np.linspace(t.ppf(0.01),\n", " t.ppf(0.99), 100)\n", "fp = t.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución t de Student')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = t.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma t de Student')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Distribución de Pareto\n", "\n", "La [Distribución de Pareto](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Pareto) esta dada por la función:\n", "\n", "$$p(x; \\alpha, k) = \\frac{\\alpha k^{\\alpha}}{x^{\\alpha + 1}} \n", "$$\n", "\n", "En dónde la variable $x \\ge k$ y el parámetro $\\alpha > 0$ son [números reales](https://es.wikipedia.org/wiki/N%C3%BAmero_real). Esta [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) fue introducida por su inventor, [Vilfredo Pareto](https://es.wikipedia.org/wiki/Vilfredo_Pareto), con el fin de explicar la distribución de los salarios en la sociedad. La [Distribución de Pareto](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_Pareto) se describe a menudo como la base de la [regla 80/20](https://es.wikipedia.org/wiki/Principio_de_Pareto). Por ejemplo, el 80% de las quejas de los clientes con respecto al funcionamiento de su vehículo por lo general surgen del 20% de los componentes." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Graficando Pareto\n", "k = 2.3 # parametro de forma.\n", "pareto = stats.pareto(k)\n", "x = np.linspace(pareto.ppf(0.01),\n", " pareto.ppf(0.99), 100)\n", "fp = pareto.pdf(x) # Función de Probabilidad\n", "plt.plot(x, fp)\n", "plt.title('Distribución de Pareto')\n", "plt.ylabel('probabilidad')\n", "plt.xlabel('valores')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histograma\n", "aleatorios = pareto.rvs(1000) # genera aleatorios\n", "cuenta, cajas, ignorar = plt.hist(aleatorios, 20)\n", "plt.ylabel('frequencia')\n", "plt.xlabel('valores')\n", "plt.title('Histograma de Pareto')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ¿Cómo elegir la distribución que mejor se ajusta a mis datos?\n", "\n", "Ahora ya tenemos un conocimiento general de las principales [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) con que nos podemos encontrar; pero ¿cómo determinamos que [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) debemos utilizar?\n", "\n", "Un modelo que podemos seguir cuando nos encontramos con datos que necesitamos ajustar a una [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad), es comenzar con los datos sin procesar y responder a cuatro preguntas básicas acerca de los mismos, que nos pueden ayudar a caracterizarlos. La **primer pregunta** se refiere a si los datos **pueden tomar valores [discretos](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad#Distribuciones_de_variable_discreta) o [continuos](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad_continua)**. **La segunda pregunta** que nos debemos hacer, hace referencia a la **[simetría](https://es.wikipedia.org/wiki/Asimetr%C3%ADa_estad%C3%ADstica) de los datos** y si hay asimetría, en qué dirección se encuentra; en otras palabras, son los [valores atípicos](https://es.wikipedia.org/wiki/Valor_at%C3%ADpico) positivos y negativos igualmente probables o es uno más probable que el otro. **La tercer pregunta** abarca los **límites superiores e inferiores en los datos**; hay algunos datos, como los ingresos, que no pueden ser inferiores a cero, mientras que hay otros, como los márgenes de operación que no puede exceder de un valor (100%). **La última pregunta** se refiere a la **posibilidad de observar valores extremos** en la [distribución](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad); en algunos casos, los valores extremos ocurren con muy poca frecuencia, mientras que en otros, se producen con mayor frecuencia.\n", "Este proceso, lo podemos resumir en el siguiente gráfico:\n", "\n", "\"Distribuciones\n", "\n", "Con la ayuda de estas preguntas fundamentales, más el conocimiento de las distintas [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) deberíamos estar en condiciones de poder caracterizar cualquier [conjunto de datos](https://es.wikipedia.org/wiki/Conjunto_de_datos)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Con esto concluyo este *tour* por las principales [distribuciones](https://es.wikipedia.org/wiki/Distribuci%C3%B3n_de_probabilidad) utilizadas en [estadística](http://relopezbriega.github.io/tag/estadistica.html). Para más información también pueden visitar mi artículo [Probabilidad y Estadística con Python](http://relopezbriega.github.io/blog/2015/06/27/probabilidad-y-estadistica-con-python/) o la categoría [estadística](http://relopezbriega.github.io/tag/estadistica.html) del blog. Espero les resulte útil.\n", "\n", "Saludos!\n", "\n", "*Este post fue escrito utilizando Jupyter notebook. Pueden descargar este [notebook](https://github.com/relopezbriega/relopezbriega.github.io/blob/master/downloads/DistStatsPy.ipynb) o ver su version estática en [nbviewer](http://nbviewer.ipython.org/github/relopezbriega/relopezbriega.github.io/blob/master/downloads/DistStatsPy.ipynb).*" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.5.1+" } }, "nbformat": 4, "nbformat_minor": 0 }