Soluciones/Práctica5.ipynb

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   "source": [
    "library(ggplot2)\n",
    "theme_set(theme_light() + theme(text=element_text(size=20)))\n",
    "\n",
    "# Funciones auxiliares\n",
    "\n",
    "eunif = function(a, b) (a + b) / 2\n",
    "vunif = function(a, b) (b - a)^2 / 12\n",
    "munif = function(a, b, t) (exp(t * b) - exp(t * a)) / t * (b - a)\n",
    "\n",
    "enorm = function(u, sd) u\n",
    "vnorm = function(u, sd) sd^2\n",
    "mnorm = function(u, sd, t) exp(sd^2 * t^2 / 2 + u * t)\n",
    "\n",
    "egamma = function(alpha, lambda) alpha / lambda\n",
    "vgamma = function(alpha, lambda) alpha / lambda^2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61f11705-d649-4ec0-aef7-773d35d2ef35",
   "metadata": {},
   "source": [
    "# Práctica 5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b305cd7-d123-4ea0-b1aa-b9c837359de4",
   "metadata": {},
   "source": [
    "## Ejercicio 1\n",
    "\n",
    "En una línea de producción los productos pasan por 4 procesos sucesivos (preparación, armado, control y embalaje) hasta quedar listos para la venta. Sean $X_1$, $X_2$, $X_3$ y $X_4$ los tiempos que tarda en cumplirse cada uno de los procesos (en minutos). Se sabe que los cuatro procesos actuán en forma independiente y que $X_1 \\sim \\mathcal{U}(3,5)$, $X_2 \\sim \\mathcal{N}(3,\\tfrac{1}{4})$, $X_3 \\sim \\Gamma(6, 3)$ y $X_4 \\sim \\mathcal{U}(2, 4)$. Sea $Y$ la variable aleatoria que mide el tiempo que tarda un producto en pasar por toda la línea de producción. Hallar $E(Y)$ y $V(Y)$.\n",
    "\n",
    "$Y = X_1 + X_2 + X_3 + X_4$\n",
    "\n",
    "$E(Y) = E(X_1 + X_2 + X_3 + X_4) = E(X_1) + E(X_2) + E(X_3) + E(X_4)$\n",
    "\n",
    "$V(Y) = V(X_1 + X_2 + X_3 + X_4) = V(X_1) + V(X_2) + V(X_3) + V(X_4)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4743b18c-4e12-4584-8448-e20ecc3161b5",
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    {
     "data": {
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       "12"
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       "12"
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       "12"
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      "text/plain": [
       "[1] 12"
      ]
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   ],
   "source": [
    "# E(Y)\n",
    "eunif(3, 5) + enorm(3, sqrt(1/4)) + egamma(6, 3) + eunif(2, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7bcb9744-2557-4d32-96b3-0450cc048ae5",
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   "outputs": [
    {
     "data": {
      "text/html": [
       "1.58333333333333"
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      "text/latex": [
       "1.58333333333333"
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       "1.58333333333333"
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       "[1] 1.583333"
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   ],
   "source": [
    "# V(Y)\n",
    "vunif(3, 5) + vnorm(3, sqrt(1/4)) + vgamma(6, 3) + vunif(2, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db0eca9b-c6bc-4ab3-bac5-fafa4feb9def",
   "metadata": {},
   "source": [
    "# Ejercicio 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb742756-b2ae-4f26-9d84-8da1af4253cd",
   "metadata": {},
   "source": [
    "## Pregunta A\n",
    "\n",
    "Sea $X$ una v.a. con distribución desconocida tal que $E(X) = 5$ y $V(X) = 0.1$. Usando la desigualdad de Chebyshev, determinar una cota inferior para la probabilidad de que $X$ esté entre $4.5$ y $5.5$.\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "P(4.5 \\leq X \\leq 5.5)\n",
    "&= P(-0.5 \\leq X-5 \\leq 0.5) \\\\\n",
    "&= P(|X-5| \\leq 0.5) \\\\\n",
    "&= P(|X-E(X)| \\leq 0.5) \\\\\n",
    "&\\geq 1 - \\tfrac{V(X)}{0.5^2}\n",
    "= 1 - \\tfrac{0.1}{0.25}\n",
    "= 0.6\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cbc94f82-8f51-440a-92d0-afa3a6e937af",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Sea $X_1, \\dots, X_n$ una muestra aleatoria (v.a. independientes e idénticamente distribuídas), con $E(X_1) = 5$ y $V(X_1) = 0.1$, y sea $\\bar{X}$ su promedio. Determinar una cota inferior, cuando n = 10, para la probabilidad de que $\\bar{X}$ esté entre $4.5$ y $5.5$.\n",
    "\n",
    "$E(\\bar{X}) = E(\\sum_{i=1}^{n} \\tfrac{E(X_i)}{n}) = \\tfrac{1}{n} \\sum_{i=1}^{n} E(X_i) = \\tfrac{1}{n} n E(X_1) = E(X_1) = 5$\n",
    "\n",
    "$V(\\bar{X}) = V(\\sum_{i=1}^{n} \\tfrac{V(X_i)}{n}) = \\tfrac{1}{n^2} \\sum_{i=1}^{n} V(X_i) = \\tfrac{1}{n^2} n V(X_1) = \\tfrac{V(X_1)}{n} = \\tfrac{0.1}{10} = 0.01$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "P(4.5 \\leq \\bar{X} \\leq 5.5)\n",
    "&= 1 - P(|\\bar{X}-5| \\geq 0.5) \\\\\n",
    "&= 1 - P(|\\bar{X}-E(\\bar{X})| \\geq 0.5) \\\\\n",
    "&\\geq 1 - \\tfrac{V(\\bar{X})}{0.5^2}\n",
    "= 1 - \\tfrac{0.01}{0.25}\n",
    "= 0.96\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac305970-d3e3-44a4-bd44-32e82d57307a",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "¿Qué pasa con la cota hallada en (b) cuando el tamaño de la muestra $n$ tiende a infinito?\n",
    "\n",
    "La cota tiende a 1, y en efecto $P(4.5 \\leq \\bar{X} \\leq 5.5) \\to 1$. Lo cual tiene sentido intuitivamente. Si $\\bar{X}$ representa el promedio de todas las $X_i$, tomando una muestra infinitamente grande, este promedio tiende al valor esperado de cada $X_i$, el cual es $E(X_1) = 5$ (todas las $X_i$ son i.i.d.).\n",
    "\n",
    "Igual hagamos las cuentas:\n",
    "\n",
    "$E(\\bar{X}) = 5$ $\\forall n$\n",
    "\n",
    "$V(\\bar{X}) = \\tfrac{V(X_1)}{n}$\n",
    "\n",
    "$P(4.5 \\leq \\bar{X} \\leq 5.5) \\geq 1 - \\tfrac{V(\\bar{X})}{0.5^2} \\geq 1 - \\tfrac{V(X_1)}{0.25n} \\geq 1 - \\tfrac{0.1}{0.25n}$\n",
    "\n",
    "Observemos que:\n",
    "\n",
    "$\\lim_{n \\to \\infty} \\tfrac{0.1}{0.25n} = 0$\n",
    "\n",
    "Luego:\n",
    "\n",
    "$\\lim_{n \\to \\infty} P(4.5 \\leq \\bar{X} \\leq 5.5) \\geq \\lim_{n \\to \\infty} 1 - \\tfrac{0.1}{0.25n} = 1 \\implies P(4.5 \\leq \\bar{X} \\leq 5.5) = 1$ cuando $n \\to \\infty$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29ed98dd-0d3f-4187-8ce5-95c60420a936",
   "metadata": {},
   "source": [
    "## Ejercicio 3\n",
    "\n",
    "El número de aviones que aterrizan en un aeropuerto sigue un proceso de Poisson con intensidad de 10 aviones cada 20 minutos. Determinar una cota inferior para la probabilidad de que el número de aviones que aterrizan en un período de 1 hora esté entre 20 y 40.\n",
    "\n",
    "$X$ = cantidad de aviones cada 1 hora\n",
    "\n",
    "$X \\sim P(30)$\n",
    "\n",
    "$E(X) = 30$\n",
    "\n",
    "$V(X) = 30$\n",
    "\n",
    "$P(20 \\leq X \\leq 40) = P(20-30 \\leq X-30 \\leq 40-30) = P(|X-E(X)| \\leq 10) \\geq 1 - \\tfrac{Var(X)}{10^2} = 1 - \\tfrac{30}{100} = 0.7$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9d88cee-7633-4105-898b-86ea23cf8e80",
   "metadata": {},
   "source": [
    "## Ejercicio 4\n",
    "\n",
    "Sea $p$ la probabilidad de que una persona elegida al azar apoye la pena de muerte. Para estimar $p$ se encuesta a $n$ personas, se cuenta la cantidad de ellas que apoya la pena de muerte ($X$) y se define la frecuencia relativa:\n",
    "\n",
    "$f_n = \\tfrac{X}{n}$\n",
    "\n",
    "Cuanto más cerca esté $f_n$ de $p$, mejor estimador será."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "517e445b-6342-48e7-8167-f351ae9c1e70",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Hallar una cota superior para $P(|f_n-p| > 0.1)$ que no dependa de $p$. ¿Cómo se puede mejorar la estimación?\n",
    "\n",
    "$X \\sim Bi(n, p)$\n",
    "\n",
    "$E(X) = np$\n",
    "\n",
    "$V(X) = np(1-p)$\n",
    "\n",
    "$E(f_n) = E(\\tfrac{X}{n}) = \\tfrac{E(X)}{n} = \\tfrac{np}{n} = p$\n",
    "\n",
    "$V(f_n) = V(\\tfrac{X}{n}) = \\tfrac{V(X)}{n^2} = \\tfrac{np(1-p)}{n^2} = \\tfrac{p(1-p)}{n}$\n",
    "\n",
    "Por Chebyshev:\n",
    "\n",
    "$P(|f_n-p| > 0.1) \\leq \\tfrac{V(f_n)}{0.1^2} = \\tfrac{p(1-p)}{0.01n}$\n",
    "\n",
    "Observemos que $p(1-p)$ describe una parábola \"triste\" con vértice en $(\\tfrac{1}{2}, \\tfrac{1}{4}) \\implies p(1-p) \\leq \\tfrac{1}{4}$ $\\forall p$"
   ]
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   "execution_count": 4,
   "id": "246ae6cb-d944-408e-be0f-d70a0e17ceea",
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HcEEEDAbwIaGOldeX2eVL/wa3CkvQM0WNLhsbULXbRzYiWSpX7u9OnTx9xVUi8d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\naf/+/bJ27dqs1pupykiQMiUboXL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AEE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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(x=c(0, 1)), aes(x=x)) +\n",
    "    stat_function(fun=function(p) p * (1-p)) +\n",
    "    geom_hline(yintercept=1/4, linetype='dashed', color='red', size=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9fe4de1-00b9-4c05-8ec4-3e21dc569c57",
   "metadata": {},
   "source": [
    "Luego, podemos acotar el numerador por $\\tfrac{1}{4}$:\n",
    "\n",
    "$P(|f_n-p| > 0.1) \\leq \\tfrac{p(1-p)}{0.01n} \\leq \\tfrac{\\tfrac{1}{4}}{0.01n} = \\tfrac{1}{0.04n} = \\tfrac{25}{n}$\n",
    "\n",
    "Para mejorar la estimación se pueden encuestar a más personas, aumentando así el $n$ y consecuentemente achicando la cota."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6a81987-9a89-4269-8235-7e86407e6720",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿A cuánta gente debería encuestarse para que $P(|f_n-p| > 0.1) \\leq 0.1$?\n",
    "\n",
    "$\\tfrac{25}{n} \\leq 0.1 \\iff \\tfrac{25}{0.1} \\leq n \\iff 250 \\leq n$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a8e3a19-b7a5-4bea-9459-1f3b87be8e88",
   "metadata": {},
   "source": [
    "## Ejercicio 6\n",
    "\n",
    "Sea $X_1, X_2, \\dots$ una sucesión de variables aleatorias independientes con distribución $P(\\lambda)$. Sea $Y_n = \\tfrac{1}{n} \\sum_{i=1}^{n} X_i^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1954782f-2ac3-4f14-8c9d-4a8edb4454aa",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Calcular $E(X_i^2)$.\n",
    "\n",
    "$M_{X_i}(t) = e^{\\lambda(e^t - 1)}$\n",
    "\n",
    "$E(X_i^2) = \\frac{\\partial^2}{\\partial t^2} M_{X_i}(t) \\Big|_{t=0} = \\lambda + \\lambda^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15bd8322-7690-4daa-834b-0993f4be5314",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "Verificar que $V(X_i^2)$ es finita. (Sugerencia: utilizar la función generadora de momentos)\n",
    "\n",
    "$E(X_i^4) = \\frac{\\partial^4}{\\partial t^4} M_{X_i}(t) \\Big|_{t=0} = \\lambda + 7 \\lambda^2 + 6 \\lambda^3 + \\lambda^4$\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "V(X_i^2)\n",
    "&= E((X_i^2)^2) - E(X_i^2)^2 \\\\\n",
    "&= E(X_i^4) - E(X_i^2)^2 \\\\\n",
    "&= \\lambda + 7 \\lambda^2 + 6 \\lambda^3 + \\lambda^4 - (\\lambda + \\lambda^2)^2 \\\\\n",
    "&= \\lambda + 7 \\lambda^2 + 6 \\lambda^3 + \\lambda^4 - (\\lambda^2 + 2\\lambda^3 + \\lambda^4) \\\\\n",
    "&= \\lambda + 6 \\lambda^2 + 4 \\lambda^3\n",
    "\\end{align*}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca706bce-4f30-4dc2-bd7a-52309b8437f6",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "Calcular el límite en probabilidad de $Y_n$.\n",
    "\n",
    "$X_i^2$ son v.a.i.i.d $\\forall i$\n",
    "\n",
    "$V(X_i^2) < \\infty$\n",
    "\n",
    "Por la Ley de los Grandes Números:\n",
    "\n",
    "$Y_n = \\tfrac{1}{n} \\sum_{i=1}^{n} X_i^2 \\xrightarrow{p} E(X_i^2) = \\lambda + \\lambda^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74168699-a988-4fb8-8784-14ed4fbd1d3f",
   "metadata": {},
   "source": [
    "## Ejercicio 7\n",
    "\n",
    "Una empresa láctea produce una cierta variedad de queso en unidades cuyo peso (en kg) es una v.a. con media 2 y varianza 0.04."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2861eebe-77b4-4bd4-b127-4a7fe76382cc",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "Calcular en forma aproximada la probabilidad de que 60 quesos pesen más de 122 kg.\n",
    "\n",
    "$X_i$ = Peso de un queso\n",
    "\n",
    "$E(X_i) = 2$\n",
    "\n",
    "$V(X_i) = 0.04$\n",
    "\n",
    "Usamos el Teorema Central del Límite (versión \"suma\").\n",
    "\n",
    "$S_n = \\sum_{i=1}^{n} X_i$ con $X_i$ v.a.i.i.d\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\\frac{S_n - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}} \\xrightarrow{d} Z \\sim \\mathcal{N}(0, 1)\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "Sea $n=60$\n",
    "\n",
    "$P(S_n > 122) \\approx P(Z > \\frac{122 - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}}) = P(Z > \\frac{122 - 60*2}{\\sqrt{60} \\sqrt{0.04}})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c056cbcc-cf57-4d17-9be5-e7c4a38cc813",
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "[1] 0.0983528"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm((122 - 60 * 2) / (sqrt(60) * sqrt(0.04)), lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4717963-e247-4898-8187-50a7e0f7d368",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿Cuántas unidades serán necesarias para satisfacer un pedido de 5000 kg. con probabilidad mayor o igual que 0.95?\n",
    "\n",
    "Intuitivamente el $n$ debería estar alrededor de 2500, pues si $E(X_i) = 2$ kg, para llegar a 5000 kg necesitamos aprox 2500 quesos."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4b47015-8893-4c09-8c82-68e85efd33dc",
   "metadata": {},
   "source": [
    "Primero buscamos el cuantil de la distribución normal estándar tal que la probabilidad sea mayor o igual que 0.95."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c516d321-1a71-4685-8454-16bc1cb2991e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-1.64485362695147"
      ],
      "text/latex": [
       "-1.64485362695147"
      ],
      "text/markdown": [
       "-1.64485362695147"
      ],
      "text/plain": [
       "[1] -1.644854"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qnorm(0.95, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23dcdf4a-efb3-43a5-a23d-0e83e6c587d0",
   "metadata": {},
   "source": [
    "$P(Z > z) = 0.95 \\iff z \\approx -1.644853$\n",
    "\n",
    "Ahora \"desestandarizamos\" este valor de $z$ para poder despejar $n$.\n",
    "\n",
    "$\\frac{5000 - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}} = z \\iff \\frac{5000 - 2n}{0.2 \\sqrt{n}} = -1.644853 \\iff n \\approx 2508.24$\n",
    "\n",
    "Como $n$ debe ser $\\mathbb{N}$ tomamos $n = 2509$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "716abf75-787f-46eb-b860-0bced8633340",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "¿En qué cambiarían los resultados obtenidos en a) y b) si el peso fuera una variable con distribución $\\mathcal{N}(2, 0.04)$?\n",
    "\n",
    "Al saber la distribución exacta los resultados serían exactos y no aproximados."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27e69ead-1376-4833-a225-e8cdc8a96620",
   "metadata": {},
   "source": [
    "## Ejercicio 8\n",
    "\n",
    "Sea $X$ una v.a. con distribución $Bi(100, 0.8)$. Usando el Teorema Central del Límite (con corrección por continuidad), calcular en forma aproximada:\n",
    "\n",
    "$E(X) = np = 100 * 0.8 = 80$\n",
    "\n",
    "$V(X) = np(1-p) = 100 * 0.8 (1 - 0.8) = 16$\n",
    "\n",
    "Por TCL, se puede aproximar $X$ con $Y \\sim \\mathcal{N}(80, 16)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "856fde8a-81ca-4642-afac-08fd9acfb697",
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 80\n",
    "sd = sqrt(16)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6f2491b-cb62-40d4-8eea-5e2fd0dd3b81",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "$P(75 \\leq X \\leq 85) \\approx P(74.5 \\leq Y \\leq 85.5)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b24e1a44-5c2e-4c4d-8e0f-50a7fbef2a1c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.830868555297329"
      ],
      "text/latex": [
       "0.830868555297329"
      ],
      "text/markdown": [
       "0.830868555297329"
      ],
      "text/plain": [
       "[1] 0.8308686"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm(85.5, mu, sd) - pnorm(74.5, mu, sd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6db37506-c06a-4891-a950-0b0cac528fbb",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "$P(X \\geq 80) \\approx P(Y \\geq 79.5)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b6fe55c7-8352-4536-9191-9ee34396995d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.549738224830113"
      ],
      "text/latex": [
       "0.549738224830113"
      ],
      "text/markdown": [
       "0.549738224830113"
      ],
      "text/plain": [
       "[1] 0.5497382"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm(79.5, mu, sd, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef338890-1592-45d8-b598-e5ec1afb4c2d",
   "metadata": {},
   "source": [
    "### Pregunta C\n",
    "\n",
    "$P(0.7 \\leq \\tfrac{X}{100} \\leq 0.8) = P(70 \\leq X \\leq 80) \\approx P(69.5 \\leq Y \\leq 80.5)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a3301510-feaf-4197-8671-8d7d5abcefe8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.5454057764671"
      ],
      "text/latex": [
       "0.5454057764671"
      ],
      "text/markdown": [
       "0.5454057764671"
      ],
      "text/plain": [
       "[1] 0.5454058"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm(80.5, mu, sd) - pnorm(69.5, mu, sd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "078fa7c6-c8a5-4619-bf48-5a640b858b42",
   "metadata": {},
   "source": [
    "### Pregunta D\n",
    "\n",
    "$P(X=80) = P(80 \\leq X \\leq 80) \\approx P(79.5 \\leq Y \\leq 80.5)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "47350427-f861-49e1-bda9-41a617925382",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0994764496602258"
      ],
      "text/latex": [
       "0.0994764496602258"
      ],
      "text/markdown": [
       "0.0994764496602258"
      ],
      "text/plain": [
       "[1] 0.09947645"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm(80.5, mu, sd) - pnorm(79.5, mu, sd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e80541e-c847-4d40-beed-65e737bf8534",
   "metadata": {},
   "source": [
    "## Ejercicio 9\n",
    "\n",
    "Para rellenar una zona baja del río llegan diariamente 3 camiones A, B y C llenos de piedras. Los pesos en toneladas de las cargas de los camiones son v.a. independientes con esperanzas 1.8, 3.8, 4.1 y varianzas 0.1, 0.18, 0.25 respectivamente.\n",
    "\n",
    "Sea $X_i$ = \"Toneladas del día i\"\n",
    "\n",
    "$E(X_i) = E(A+B+C) = E(A) + E(B) + E(C) = 1.8 + 3.8 + 4.1 = 9.7$\n",
    "\n",
    "$V(X_i) = V(A+B+C) = V(A) + V(B) + V(C) = 0.1 + 0.18 + 0.25 = 0.53$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db74f206-2a67-4f11-bd4f-a903161f0df8",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "¿Cuál es la probabilidad aproximada de que en 360 días se superen las 3500 t?\n",
    "\n",
    "Sea $S_n = \\sum_{i=0}^n X_i$ la suma total de toneladas en $n$ días. $X_i$ v.a.i.i.d.\n",
    "\n",
    "Por el TCL (versión \"suma\"):\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\\frac{S_n - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}} \\xrightarrow{d} Z \\sim \\mathcal{N}(0, 1)\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "Con $n = 360$, calculemos la probabilidad pedida:\n",
    "\n",
    "$P(S_n > 3500) \\approx P(Z > \\tfrac{3500 - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}}) = P(Z > \\tfrac{3500 - 360 * 9.7}{\\sqrt{360} \\sqrt{0.53}})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a67d70aa-62ca-428d-831b-cf8f977d35bd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.281239600247931"
      ],
      "text/latex": [
       "0.281239600247931"
      ],
      "text/markdown": [
       "0.281239600247931"
      ],
      "text/plain": [
       "[1] 0.2812396"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm((3500 - 360 * 9.7) / (sqrt(360) * sqrt(0.53)), lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98535d19-1197-4af3-aacf-d60259370995",
   "metadata": {
    "tags": []
   },
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿Cuánto tiempo será necesario para superar las 3500 t con una probabilidad aproximada de 0.90 por lo menos?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6678d0d8-1107-4462-80b9-41eb873d1e70",
   "metadata": {},
   "source": [
    "Primero buscamos el cuantil de la distribución normal estándar tal que la probabilidad sea mayor o igual que 0.9."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6c685fa9-372e-45bd-a1f5-727d9eab2c97",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-1.2815515655446"
      ],
      "text/latex": [
       "-1.2815515655446"
      ],
      "text/markdown": [
       "-1.2815515655446"
      ],
      "text/plain": [
       "[1] -1.281552"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qnorm(0.9, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74ed63a8-0131-4ed1-b660-af21a44be41c",
   "metadata": {},
   "source": [
    "$P(Z > z) = 0.9 \\iff z \\approx -1.281551$\n",
    "\n",
    "Ahora \"desestandarizamos\" este valor de $z$ para poder despejar $n$.\n",
    "\n",
    "$\\frac{3500 - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}} = z \\iff \\frac{3500 - 9.7n}{\\sqrt{n}\\sqrt{0.53}} = -1.281551 \\iff n \\approx 362.656$\n",
    "\n",
    "Como $n$ debe ser $\\mathbb{N}$ tomamos $n = 363$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1cea8809-680b-4627-b7fe-1ae37f519410",
   "metadata": {},
   "source": [
    "## Ejercicio 10\n",
    "\n",
    "Sean $X_1, X_2, \\dots, X_n$ v.a. independientes e idénticamente distribuidas con función de densidad:\n",
    "\n",
    "$f_X(x) = \\tfrac{2}{x^3} I_{[1, \\infty)}(x)$\n",
    "\n",
    "Calcular aproximadamente:\n",
    "\n",
    "$P(\\prod_{i=1}^{100} X_i > e^{55})$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9c55382-20bf-4b97-973b-f5cae195a6df",
   "metadata": {},
   "source": [
    "Sale truco de la galera:\n",
    "\n",
    "Sea $Y = ln(X)$\n",
    "\n",
    "$F_Y(y) = P(Y \\leq y) = P(ln(X) \\leq y) = P(e^{ln(X)} \\leq e^y) = P(X \\leq e^y) = F_X(e^y)$\n",
    "\n",
    "$\\tfrac{\\partial}{\\partial y} F_Y(y) = \\tfrac{\\partial}{\\partial y} F_X(e^y) \\iff f_Y(y) = f_X(e^y) e^y = \\tfrac{2}{e^{3y}} e^y = 2e^{-3y} e^y = 2e^{-2y}$\n",
    "\n",
    "Luego:\n",
    "\n",
    "$f_Y(y) = 2e^{-2y} I_{[1, \\infty)}(y) \\implies Y \\sim \\mathcal{E}(2)$\n",
    "\n",
    "$E(Y) = \\tfrac{1}{2}$\n",
    "\n",
    "$V(Y) = \\tfrac{1}{4}$\n",
    "\n",
    "Sigamos usando las propiedades del logaritmo, puntualmente vamos a usar:\n",
    "\n",
    "$ln(e^x) = x$\n",
    "\n",
    "$ln(A*B) = ln(A) + ln(B)$\n",
    "\n",
    "$P(\\prod_{i=1}^{100} X_i > e^{55}) = P(ln( \\prod_{i=1}^{100} X_i ) > ln(e^{55})) = P(\\sum_{i=1}^{100} ln(X_i) > 55) = P(\\sum_{i=1}^{100} Y_i > 55)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a289b9fe-f183-4203-8fe9-8a636312cb8f",
   "metadata": {},
   "source": [
    "### Cálculo aproximado\n",
    "\n",
    "Sea $Z \\sim \\mathcal{N}(0, 1)$\n",
    "\n",
    "Por el TCL:\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "P\\Big(\\sum_{i=1}^{100} Y_i > 55 \\Big)\n",
    "&= P\\Big(Z > \\frac{55 - 100 E(Y_i)}{\\sqrt{100} \\sqrt{V(Y_i)}} \\Big) \\\\\n",
    "&= P\\Big(Z > \\frac{55 - 100 \\tfrac{1}{2}}{10 \\tfrac{1}{2}} \\Big) \\\\\n",
    "&= P(Z > 1)\n",
    "\\end{align*}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "d4c23d55-1c8a-4ed3-861e-b973d66401fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.158655253931457"
      ],
      "text/latex": [
       "0.158655253931457"
      ],
      "text/markdown": [
       "0.158655253931457"
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      "text/plain": [
       "[1] 0.1586553"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pnorm(1, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01d3c801-e020-4bec-9a63-e252585cc946",
   "metadata": {},
   "source": [
    "### Cálculo exacto\n",
    "\n",
    "Hay una propiedad que podemos usar para calcular la probabilidad exacta.\n",
    "\n",
    "Sea $Z = \\sum_{i=1}^{100} Y_i$ con $Y_i \\sim \\mathcal{E}(2)$ v.a.i.i.d. $\\implies Z \\sim \\Gamma(100, 2)$\n",
    "\n",
    "$P(Z > 55)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "3adcb37e-3868-485c-aa70-3f7e041d4c2f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.158278670060087"
      ],
      "text/latex": [
       "0.158278670060087"
      ],
      "text/markdown": [
       "0.158278670060087"
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      "text/plain": [
       "[1] 0.1582787"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pgamma(55, 100, 2, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7218236b-65e4-4a19-bf15-2c3c1119ea27",
   "metadata": {},
   "source": [
    "## Ejercicio 11\n",
    "\n",
    "En cierto juego de azar la probabilidad de ganar es 0.3. Para participar en el mismo se paga un peso y, en caso de ganar, se reciben 5 pesos."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4e7dee4-d809-42b8-916e-6aa981087cb1",
   "metadata": {},
   "source": [
    "### Pregunta A\n",
    "\n",
    "¿Cuál es la probabilidad aproximada de que en 100 juegos independientes un jugador gane más de 90 pesos?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e043de79-0075-4b2e-a59b-ab742d26f64b",
   "metadata": {},
   "source": [
    "$X$ = Ganancia de un juego\n",
    "\n",
    "$P(X=4) = 0.3$\n",
    "\n",
    "$P(X=-1) = 0.7$\n",
    "\n",
    "$E(X) = 4 * 0.3 + (-1) * 0.7 = 0.5$\n",
    "\n",
    "$V(X) = E(X^2) - E(X)^2 = (4^2 * 0.3 + (-1)^2 * 0.7) - 0.5^2 = 5.25$\n",
    "\n",
    "Por el TCL:\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "P\\Big(\\sum_{i=1}^{100} X_i > 90 \\Big) = P\\Big(Z > \\frac{90 - 100 E(X_i)}{\\sqrt{100} \\sqrt{V(X_i)}} \\Big) = P\\Big(Z > \\frac{90 - 100 * 0.5}{10 \\sqrt{5.25}} \\Big)\n",
    "\\end{align*}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "08c5a545-6d25-4789-8940-3ace27dbb0b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0404277991850261"
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      "text/latex": [
       "0.0404277991850261"
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       "0.0404277991850261"
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      "text/plain": [
       "[1] 0.0404278"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = (90 - 100 * 0.5) / (10 * sqrt(5.25))\n",
    "pnorm(z, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14827451-4bae-4827-bd2e-66cac3b4bc46",
   "metadata": {},
   "source": [
    "**Otra forma para calcularlo más preciso podría ser así**\n",
    "\n",
    "$X$ = Cantidad de juegos ganados\n",
    "\n",
    "$X \\sim Bi(100, 0.3)$\n",
    "\n",
    "$Y$ = Ganancia\n",
    "\n",
    "$Y = X * 4 - (100 - X) = 5X - 100$\n",
    "\n",
    "$P(Y > 90) = P(5X - 100 > 90) = P(X > 38)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0efe499d-41fc-49ad-a21b-c8456b020c5e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0339789982689313"
      ],
      "text/latex": [
       "0.0339789982689313"
      ],
      "text/markdown": [
       "0.0339789982689313"
      ],
      "text/plain": [
       "[1] 0.033979"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pbinom(38, 100, 0.3, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dabafb02-05a0-49d5-8923-38b9304b4e78",
   "metadata": {},
   "source": [
    "### Pregunta B\n",
    "\n",
    "¿Cuántas veces tendrá que jugar para ganar más de 90 pesos con probabilidad aproximada mayor o igual que 0.8?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8256638-5340-4ad5-8521-7e88ece07e00",
   "metadata": {},
   "source": [
    "Primero buscamos el cuantil de la distribución normal estándar tal que la probabilidad sea mayor o igual que 0.8."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "16f2d07c-72db-4ed2-83d2-28989b34049b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-0.841621233572914"
      ],
      "text/latex": [
       "-0.841621233572914"
      ],
      "text/markdown": [
       "-0.841621233572914"
      ],
      "text/plain": [
       "[1] -0.8416212"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qnorm(0.8, lower.tail=FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50a77e09-7afa-408e-891c-c69a8aa06fc5",
   "metadata": {},
   "source": [
    "$P(Z > z) = 0.8 \\iff z \\approx -0.841621$\n",
    "\n",
    "Ahora \"desestandarizamos\" este valor de $z$ para poder despejar $n$.\n",
    "\n",
    "$\\frac{90 - n E(X_i)}{\\sqrt{n} \\sqrt{V(X_i)}} = z \\iff \\frac{90 - 0.5n}{\\sqrt{n}\\sqrt{5.25}} = -0.841621 \\iff n \\approx 239.713$\n",
    "\n",
    "Como $n$ debe ser $\\mathbb{N}$ tomamos $n = 240$"
   ]
  }
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