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"# Variables Aleatorias\n",
"\n",
"\n",
"- Variables aleatorias\n",
"- Probabilidades puntuales, funciones de densidad y funciones de distribución acumuladas. \n",
"\n",
"\n",
"Una variable aleatoria es una función que asigna un número a cada elemento en el espacio muestral de un experimento aleatorio. \n",
"Se denomina Variable Aleatoria Discreta si los valores son enteros:\n",
"\n",
"$\\begin{array}{cc} X:& \\Omega \\to \\mathbb{N}\\\\\n",
"&\\omega \\to X(\\omega)\n",
"\\end{array}$\n",
"\n",
"y Variable Aleatoria Continua si los valores son reales:\n",
"\n",
"$\\begin{array}{cc} X:& \\Omega \\to \\mathbb{R}\\\\\n",
"&\\omega \\to X(\\omega)\n",
"\\end{array}$\n",
"\n",
"\n",
"Ejemplos: \n",
"\n",
"$\\begin{array}{ll} \n",
"X:& \\Omega \\to \\{0,1\\}\\\\\n",
"&\\omega \\to X(\\omega)= \n",
"\\left\\{\\begin{array}{ll} 1 & \\omega=cara\\\\\n",
" 0 & \\omega=sello \\\\\n",
"\\end{array} \\right .\\\\\n",
"&\\\\\n",
" X:& \\Omega \\to [0,\\infty) \\\\\n",
"&\\omega \\to X(\\omega)= t \\\\\n",
"\\end{array}$\n",
"\n",
"donde $t$ es el tiempo de falla de una máquina y $\\omega$ es el evento de falla con todas sus características no necesariamente observables."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Función de probabilidad puntual\n",
"En el caso de variables aleatorias discretas, la probabilidades se definen mediante una función de probabilidad puntual.\n",
"\n",
"Ejemplo: \n",
"\n",
"$\\begin{array}{ll} \n",
"X:& \\Omega \\to \\mathbb{N}\\\\\n",
"&\\omega \\to X(\\omega)= n \\\\\n",
"\\end{array}$\n",
"\n",
"dónde n representa el número de clientes que llegan al banco el lunes entre 9 y 10 am \n",
"\n",
"$$P(X=10) = f(10) = ?$$\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
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"source": [
"##caso en que el nro de clientes se modela como Poisson de media 10\n",
"library(rbokeh)\n",
"vec <- seq(0,30,1)\n",
"pvec <- dpois(vec,10)\n",
"p <- figure(plot_width=600,plot_height=300, title=\"Ejemplo probabilidad puntual Poisson\", title_location=\"above\") %>%\n",
" ly_points(vec,pvec, hover = list(vec,pvec))%>%\n",
" ly_segments(vec,rep(0,30),vec,pvec)\n",
"p"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Función de densidad de probabilidad\n",
"En el caso de variables aleatorias continuas, las probabilidades se definen mediante una función de densidad de probabilidad:\n",
"\n",
"$\\begin{array}{ll} \n",
"f_X:& \\mathbb{R} \\to \\mathbb{R_+}\\\\\n",
"&x \\to f_X(x) \\\\\n",
"\\end{array}$\n",
"\n",
"Ejemplo: \n",
"\n",
"$\\begin{array}{ll} \n",
"X:& \\Omega \\to \\mathbb{R}\\\\\n",
"&\\omega \\to X(\\omega)= x \\\\\n",
"f_X(x)& = \\frac{1}{2\\sqrt{2\\pi}} exp(\\frac{-(x-60)^2}{8})\n",
"\\end{array}$\n",
"\n",
"dónde x representa el peso de un individuo.\n",
"\n",
"$$P(58 <= X <= 62) = \\int_{58}^{62} f(x) dx = F_X(62) - F_X(58) = \\int_0^{62} f(x)dx - \\int_0^{58} f(x)dx ?$$\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
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"[1] 0.6826895\n"
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],
"source": [
"library(rbokeh)\n",
"vec <- seq(50,70,by=0.1)\n",
"vec2 <- seq(58,62,by=0.1)\n",
"pvec <- dnorm(vec,60,2)\n",
"par(cex=0.8)\n",
"p <- figure(plot_width=600,plot_height=300, title=\"Ejemplo densidad Normal\", title_location=\"above\") %>%\n",
" ly_lines(vec,pvec) %>%\n",
" ly_abline( h=0) %>%\n",
" ly_polygons( x = c(58,vec2,62), y =c(0,dnorm(vec2,60,2),0), col = \"grey\")\n",
"p\n",
"\n",
"\n",
"prob = pnorm(62,60,2)-pnorm(58,60,2)\n",
"print(prob)\n",
"\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Función de distribución de probabilidad acumulada\n",
"Es la función $F_X$ que devuelve la probabilidad de que una v.a. sea menor o igual que un valor.\n",
"\n",
"En el caso de las v.a. discretas se calcula como:\n",
"\n",
"$\\begin{array}{ll} \n",
"P(X \\leq x) = F_X(x) & = \\sum_{t=1}^x \\limits P(X = t)\n",
"\\end{array}$\n",
"\n",
"\n",
"En el caso de las v.a. continuas se calcula como:\n",
"\n",
"$\\begin{array}{llll} \n",
"P(X \\leq x) = &F_X(x)& = &\\int_{-\\infty}^x f(x)dx\\\\\n",
"&f_X(x)& = & \\frac{d}{dx} F_X(x)\n",
"\\end{array}$\n",
"\n",
"Y una manera alternativa de obtener la probabilidad:\n",
"\n",
"$\\begin{array}{l}\n",
"P(a \\leq X \\leq b) = F_X(b) - F_X(a)\\\\\n",
"\\end{array}$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
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"source": [
"library(rbokeh)\n",
"##distribución Poisson acumulada\n",
"vec <- seq(1,30,by=1)\n",
"pvec <- ppois(vec,10)\n",
"p1 <- figure(plot_width=600,plot_height=200, title=\"Prob. Acumulada: Caso Discreto\", title_location=\"above\") %>%\n",
" ly_points(vec,pvec, hover = list(vec,pvec))%>%\n",
" ly_segments(vec,rep(0,30),vec,pvec)\n",
"p1"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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"source": [
"##distribución normal acumulada\n",
"vec <- seq(40,80,by=0.1)\n",
"pvec1 <- pnorm(vec,60,2)\n",
"pvec2 <- pnorm(vec,60,8)\n",
"pvec3 <- pnorm(vec,60,4)\n",
"p2 <- figure(plot_width=600,plot_height=200, title=\"Prob. Acumulada: Caso Continuo\", title_location=\"above\", legend_location = \"bottom_right\") %>%\n",
" ly_lines(vec,pvec1,legend=\"sd=2\") %>%\n",
" ly_lines(vec,pvec2,col=\"blue\",legend=\"sd=8\") %>%\n",
" ly_lines(vec,pvec3,col=\"red\",legend=\"sd=4\") \n",
"p2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Variables Aleatorias Multidimensionales\n",
"Se trata de funciones que asignan un vector multidimensional al resultado de un experimento aleatorio. \n",
"Se denomina Vector Aleatorio Discreto si los valores son enteros multidimensionales:\n",
"\n",
"$\\begin{array}{cl} {\\bf X}:& \\Omega \\to \\mathbb N^k\\\\\n",
"&\\omega \\to {\\bf X}(\\omega)=(X_1(\\omega),\\cdots,X_k(\\omega)) = (n_1,\\cdots,n_k)\n",
"\\end{array}$\n",
"\n",
"y Vector Aleatorio Continuo si los valores son reales multidimensionales:\n",
"\n",
"$\\begin{array}{cl} {\\bf X}:& \\Omega \\to \\mathbb R^k\\\\\n",
"&\\omega \\to {\\bf X}(\\omega)=(X_1(\\omega),\\cdots,X_k(\\omega)) = (x_1,\\cdots,x_k)\n",
"\\end{array}$\n",
"\n",
"**Ejemplo:**\n",
"\n",
"Sea \n",
"\n",
"$\\begin{array}{cl}\n",
"(X,Y):&\\Omega \\to \\mathbb R_{+}^2\\\\\n",
"&\\omega \\to (X(\\omega),Y(\\omega))=(x,y)\\\\\n",
"\\end{array}$\n",
"\n",
"tal que su densidad de probabilidad es:\n",
"\n",
"$$f(x,y) = ab \\exp^{-(ax+by)} \\qquad a,b>0$$\n",
"\n",
"y entonces la distribución de probabilidad acumulada queda:\n",
"\n",
"$$F(x,y)= \\int_0^x \\int_0^y f(x,y) dydx\n",
"= (1- \\exp^{-ax})(1- \\exp^{-by})$$\n",
"\n",
"Notar que \n",
"\n",
"$$\\lim_{x,y \\to \\infty} F(x,y) = 1$$\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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HBxKynFKgHJ2JUqKu3YMKwUWH2/Hg29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\nTkSuQmCC",
"text/plain": [
"Plot with title \"Distribución Bivariada: Caso a=5 y b=2\""
]
},
"metadata": {
"image/png": {
"height": 300,
"width": 600
}
},
"output_type": "display_data"
}
],
"source": [
"options(repr.plot.width=10, repr.plot.height=5)\n",
"par(mfrow=c(1,2))\n",
"x <- seq(0,1,by=0.05)\n",
"y <- seq(0,1,by=0.05)\n",
"a=5\n",
"b=1\n",
"dens <- function(x,y) a*b*(exp (-a*x -b*y))\n",
"z = outer(x,y,dens)\n",
"persp(x,y,z,ticktype = \"detailed\",zlab=\"f(x,y)\",main=\"Densidad de Prob. Bivariada: Caso a=5 y b=2\")\n",
"\n",
"dist <- function(x,y) 1-exp (-a*x -b*y) \n",
"z = outer(x,y,dist)\n",
"persp(x,y,z,ticktype = \"detailed\",zlab=\"F(x,y)\",main=\"Distribución Bivariada: Caso a=5 y b=2\")\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Regla de Bayes\n",
"En su formulación con variables aleatorias, la regla de Bayes queda:\n",
"\n",
"\n",
"$\\begin{array}{lll}\n",
"P(y \\mid x)& = & \\frac{P(x \\mid y) P_Y(y)}{P_X(x)} \\,= \\, \\frac{P(x \\mid y) P_Y(y)}{\\sum_y P(x\\mid y) P_y(y))}\\\\\n",
"\\end{array}$\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"celltoolbar": "Slideshow",
"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": "4.1.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}