6109-Proba: Ejercicios Obligatorios de Estadística, en Python

$ejs.pdf

clientes_akaike.txt

41.3431
33.5864
155.0215
23.0177
6.4319
121.5841
45.4330
69.9679
0.8175
27.2983
75.4275
111.1169
33.3828
10.3938
9.7858
10.0267
87.5067
33.4421
7.3478
116.1580
8.5291
5.8085
83.8134
141.0322
41.0352
9.9160
41.5973
48.5345
37.8218
114.1760
111.2537
7.6294
27.8256
72.9788
7.9890
156.4985
42.9783
17.8992
59.4967
50.2285
35.2326
6.1482
13.7811
146.7584
70.2524
53.3084
7.9972
47.7177
78.5195
28.3108
40.4366
5.8837
3.7136
11.0567
18.9215
16.3219
5.8126
150.1327
60.0000
299.2559
10.3745
37.3112
55.2573
81.6417
34.4192
68.1809
147.5544
135.5997
147.5797
49.0625
115.0861
44.7781
5.8597
57.1713
116.5329
31.0024
7.3422
57.9646
174.4583
107.2246
141.3088
16.5217
43.1261
22.1702
19.6764
30.7857
25.5261
7.1708
22.6577
7.3182
41.8176
107.6138
147.7450
18.5093
64.7735
21.9308
23.4886
34.8096
18.4428
37.5965

ejs.tex

\documentclass[12pt]{article}

\usepackage{tikz}
\usepackage{xcolor}
\usepackage{pdfpages}
\pagestyle{empty}

\usepackage[a4paper, margin=0.2in]{geometry}

\begin{document}
\includepdf[pagecommand={
	\begin{tikzpicture}[remember picture, overlay]
		\node at (15.65,-6.5) {Vera (Curso 21)};
		\node at (15.3, -7.4) {del Mazo, Federico};
		\node at (16.4,-8.2) {100029};
	\end{tikzpicture}}]{enunciado}
\includepdf[pages={2}]{enunciado}

% jupyter nbconvert --to pdf --execute notebook.ipynb
\includepdf[pages=-]{notebook}

\end{document}

enunciado.pdf

notebook.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d0a6fce9",
   "metadata": {},
   "source": [
    "# Estimadores de Máxima Verosimilitud"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56ec373f",
   "metadata": {},
   "source": [
    "Queremos hallar el EMV de $\\theta = (\\mu, \\sigma^2)$ de la muestra $X_1,...,X_n \\sim N(\\mu,\\sigma^2)$\n",
    "\n",
    "Entonces, con muchísima ayuda de Wolfram, y justificando el método de igualar la derivada a 0 porque se que la distribución Normal es parte de la familia regular...\n",
    "\n",
    "$$ f_X(x_i) = \\frac{1}{\\sigma^2\\sqrt{2\\pi}} * e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} $$\n",
    "\n",
    "$$ f_X(\\underline{x}) = \\prod_{i=1}^{n} f_X(x_i) $$\n",
    "\n",
    "---\n",
    "\n",
    "$$ L(\\theta) = \\frac{1}{\\left(\\sqrt{2\\pi}\\right)^{n}} \\frac{1}{\\left({\\sigma^2}\\right)^n} * e^{\\frac{1}{\\sigma^2} * \\sum_{i=1}^{n} (x_i - \\mu)^2}$$ \n",
    "\n",
    "$$ \\ln(L) = -n \\ln\\left(\\sqrt{2\\pi}\\right) - n \\ln\\left(\\sigma^2\\right) - \\frac{1}{\\sigma^2} \\sum_{i=1}^{n} (x_i - \\mu)^2$$ \n",
    "\n",
    "---\n",
    "\n",
    "$$ \\frac{\\partial\\ln(L)}{\\partial{\\mu}} = \\frac{1}{\\sigma^2} \\sum_{i=1}^{n} (x_i-\\mu) = 0 \\Rightarrow \\hat\\mu = \\frac{1}{n} \\sum_{i=1}^{n} x_i$$\n",
    "\n",
    "$$ \\frac{\\partial\\ln(L)}{\\partial{\\sigma^2}} = - \\frac{n}{2\\sigma^2} + \\frac{\\sum_{i=1}^{n}(X_i - \\mu)^2}{2\\sigma^2} \\Rightarrow \\hat{\\sigma}^2 = \\frac{1}{n} \\sum_{i=1}^{n} (X_i - \\mu)^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c40321c",
   "metadata": {},
   "source": [
    "Podemos ver que, de la familia de estimadores de varianza dada en el enunciado, en el EMV tenemos que $a_n$ es $\\frac{1}{n}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e9241e8",
   "metadata": {},
   "source": [
    "$$ $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7437e95",
   "metadata": {},
   "source": [
    "Ahora, buscamos calcular el sesgo y la varianza de $T^2$\n",
    "\n",
    "$$ T^2 = \\theta = a_n \\sum_{i=1}^{n} (X_i - \\bar{X})^2$$\n",
    "\n",
    "$$ Sesgo = B_{\\theta}(\\hat{\\theta}) = E_{\\theta}[\\hat{\\theta}] - \\theta $$\n",
    "\n",
    "\n",
    "\n",
    "El teorema de Cochran nos dice que, con $X\\sim N(0,1)$, tenemos $\\sum_{i=1}^{n} (X_i - \\bar{X})^2 \\sim \\chi_{n-1}^2$. \n",
    "\n",
    "Nosotros tenemos $X\\sim N(0,\\sigma^2)$, por lo que nos queda $\\sum_{i=1}^{n} (X_i - \\bar{X})^2 \\sim \\sigma^2 \\chi_{n-1}^2$\n",
    "\n",
    "Finalmente, recordando que si $X\\sim\\chi_{k}^2$ su esperanza es $k$ y su varianza es $2k$, nos queda\n",
    "\n",
    "\n",
    "$$ B_{\\theta}(T^2) = a_n * \\sigma^2 * (n-1) - T^2 $$\n",
    "\n",
    "$$ Var_{\\theta}(T^2) = a_n^2 * \\sigma^4 * 2 * (n-1) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7daf7667",
   "metadata": {},
   "source": [
    "$$ $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6de5baa",
   "metadata": {},
   "source": [
    "Ya sabiendo el sesgo en función de $a_n$, queremos determinar el $a_n$ para que el estimador de $T^2$ sea insesgado. Un estimador es insesgado si la esperanza es igual al parametro que estamos buscando estimar.\n",
    "\n",
    "Entonces la pregunta es... ¿cómo logramos que $E[T^2]$ sea la $\\sigma^2$?\n",
    "\n",
    "$$ a_n * \\sigma^2 * (n-1) = \\sigma^2 $$\n",
    "\n",
    "$$ a_n = \\frac{1}{n-1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff0c9e84",
   "metadata": {},
   "source": [
    "$$ $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0a71757",
   "metadata": {},
   "source": [
    "Finalmente, buscamos el $a_n$ para minimizar el error cuadrático medio\n",
    "\n",
    "$$ ECM_{\\theta}(T^2) = Var_{\\theta}(T^2) + B_{\\theta}(T^2)^2 $$\n",
    "\n",
    "$$ a_n^2 * \\sigma^4 * 2 * (n-1) + (a_n * \\sigma^2 * (n-1) - T^2)^2 $$\n",
    "\n",
    "La derivada respecto de $a_n$ igualada a $0$ nos devuelve que el $a_n$ que minimiza el ECM es $\\frac{1}{n+1}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed656f6a",
   "metadata": {},
   "source": [
    "# AKAIKE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "568eb2c2",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from statsmodels.distributions.empirical_distribution import ECDF\n",
    "\n",
    "with open('clientes_akaike.txt') as f:\n",
    "        data = f.readlines()\n",
    "\n",
    "data = [float(d.strip()) for d in data]\n",
    "ecdf = ECDF(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e007edb6",
   "metadata": {},
   "outputs": [
    {
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SyjdLKZtKKX+zx2eeXUq5q/WZvy6llBH5jgAAAAAA6AhPbNiWnQM1p8yc1nQUAAAAoEPtt0BXSulN8oEkFyU5M8kbSyln7nHY5UnW1lpPTfL+JO9rbd+W5A+SvHsvp/7bJG9LMq/168ID+QYAAAAAAOhMD63enCQ58cgpDScBAAAAOtVQHuF6XpLFtdYlSVJKuTbJJUnu3e2YS5L8Yev1Z5P8TSml1Fo3J/lOKeXU3U9YSjk2yaG11u+33v9jkp9N8qUD/1YYS3OuvL7pCByA0fzvtvSqi0ft3AAAAMBPKqVcmOSvkvQm+Uit9ap9HPfa7JrZ/VStdcEYRoSD9siaLUkU6AAAAIDRM5RHuB6f5JHd3i9rbdvrMbXW/iTrk8zYzzmX7eecSZJSyhWllAWllAUrV64cQlwAAAAA6GxDfGpESinTk/x6kpvGNiGMjIfXbElfT8mxh01uOgoAAADQoYZSoGtUrfXDtdZza63nzpw5s+k4AAAAADAe/PCpEbXWHUmefGrEnv4oyfuSbBvLcDAS7n1sQ26464mccOSU9PWO+1E2AAAA0KaGMnV4NMkJu72f3dq212NKKX1JDkuyej/nnL2fcwIAAAAAe7ffp0aUUs5JckKt9fqnOpEnQDDebO8fyF98ZVFe/TffycZt/fnDV5/VdCQAAACggw2lQHdLknmllJNLKROTXJrkuj2OuS7JZa3Xr0vyjVpr3dcJa62PJ9lQSjm/lFKS/GKSLww7PQAAAADwE0opPUn+Mslv7e9YT4BgPLn94bV51V9/J///Nxbn1Wcfl6/95ovy4tOsSwAAAGD09O3vgFprfynlnUluTNKb5GO11ntKKe9NsqDWel2Sjyb5eCllcZI12VWyS5KUUpYmOTTJxFLKzyZ5Ra313iS/muTqJIck+VLrFwAAAACwf/t7asT0JE9L8q1d/341xyS5rpTy6lrrgjFLCUO0dceuu8597LsPZtahk/MPb/6pvPT0o5uOBQAAAHSB/RbokqTWekOSG/bY9p7dXm9L8vp9fHbOPrYvyK4hHgAAAAAwPD98akR2FecuTfJ/P7mz1ro+yVFPvi+lfCvJu5XnGI++v2R1fudf7sxDq7fk559zYq68aH6mT57QdCwAAACgSwypQAcAAAAAjB9DfGoEjGsbt+3MVV+6L5+46eGcNGNKPvW28/PcuTOajgUAAAB0GQU6AAAAAGhD+3tqxB7bXzIWmWA4Lr9mQRYsXZO3vuDk/NYrTs8hE3ubjgQAAAB0IQU6AAAAAADG1OPrt+bmB9fkt195et7x0lObjgMAAAB0sZ6mAwAAAAAA0F2+vnBFkuSVZ81qOAkAAADQ7RToAAAAAAAYU19fuDwnzZiSuTOnNR0FAAAA6HIKdAAAAAAAjJktO/rz3R+szgXzZ6WU0nQcAAAAoMsp0AEAAAAAMGa+88Cq7OgfzMvPOLrpKAAAAAAKdAAAAAAAjJ2vL1yR6ZP6cu6cI5uOAgAAAKBABwAAAADA2OgfGMxX7n0iLzvj6EzsM54GAAAAmmdCAQAAAADAmLjpwTVZu2VnLnrasU1HAQAAAEiiQAcAAAAAwBj50t2P55AJvXnxaTObjgIAAACQRIEOAAAAAIAxMDBYc+M9y/PS+TNzyMTepuMAAAAAJFGgAwAAAABgDNz60Nqs3Ljd41sBAACAcUWBDgAAAACAUbV5e3/e/9X7M6mvJy+df3TTcQAAAAB+qK/pAAAAAAAAdK41m3fkzVffkruWrcv7XvuMTJtkLA0AAACMHyYVAAAAAACMikfXbc0vfPSmLFu7NX/3pmfnFWcd03QkAAAAgB+jQAcAAAAAwIh7YPnG/OLHbs6m7f35+FvOy3NOmdF0JAAAAICfoEAHAAAAAMCIuvWhtXnL1bdkYl9PPn3Fc3PmcYc2HQkAAABgrxToAAAAAAAYMd9ctCK/8k+35phDJ+cf3/KcnDhjStORAAAAAPZJgQ4AAAAAgINWa80131uaP75+YU4/ZnqufvN5mTl9UtOxAAAAAJ6SAh0AAAAAAAdl286B/N7n7srnbn80Lz/j6Lz/587O9MkTmo4FAAAAsF8KdAAAAAAAHLBH1mzJ2//p1tz7+Ib8Py8/Le962anp6SlNxwIAAAAYEgU6AAAAAAAOyHcXr8o7P3lb+gdrPnrZuXnZ/FlNRwIAAAAYFgU6AAAAAACGpdaav/+PJbnqS/fl1KOn5UO/cG5OPmpq07EAAAAAhk2BDgAAAACAIduyoz///bN35ot3Pp6fefox+bPXPTNTJxk1AwAAAO3JVAMAAAAAgCH7H/96d2646/FcedH8/PKLTkkppelIAAAAAAdMgQ4AAAAAgCHZOTCYr96zPK9/9gl5+4vnNh0HAAAA4KD1NB0AAAAAAID2cOtDa7Nxe39eOv/opqMAAAAAjAgFOgAAAAAAhuSb963IhN6SF8w7qukoAAAAACNCgQ4AAAAAgCH55qIVOe/kIzNtUl/TUQAAAABGhAIdAAAAAAD7tWztlty/fFNeerrHtwIAAACdQ4EOAAAAAID9+vf7VyZJXqJABwAAAHQQBToAAAAAAPbr1qVrc9S0iZk7c2rTUQAAAABGjAIdAAAAAAD7ddvDa/OsE49IKaXpKAAAAAAjRoEOAAAAAICntHrT9ixdvSXnnHhE01EAAAAARpQCHQAAAAAAT+n2h9clSc458fBGcwAAAACMNAU6AAAAAACe0m0Pr01fT8kzZh/edBQAAACAEdXXdACApzLnyutH5bxLr7p4VM4LAAAA0IkWLF2bM449NIdM7G06CgAAAMCIcgc6AAAAAAD2ad2WHbn14bV50WlHNR0FAAAAYMQp0AEAAAAAsE/fWrQyA4M1F5wxq+koAAAAACNOgQ4AAAAAgH362sLlOWraxJw9+/CmowAAAACMOAU6AAAAAAD2aufAYP79/pV52fyj09NTmo4DAAAAMOIU6AAAAAAA2KtbHlyTjdv6Pb4VAAAA6FgKdAAAAAAA7NW/P7AyE3t78sJ5RzUdBQAAAGBUKNABAAAAALBXP1ixKafMnJopE/uajgIAAAAwKhToAAAAAADYqyWrNufko6Y2HQMAAABg1CjQAQAAAADwE/oHBvPw6i2Zo0AHAAAAdDAFOgAAAAAAfsKytVvTP1jdgQ4AAADoaAp0AAAAAAD8hAdXbU6SnKJABwAAAHQwBToAAAAAAH7MY+u25m++uTg9JTll5rSm4wAAAACMmr6mAwAAAAAAMH7ccNfjufJf7szAYM1fvuHsHDl1YtORAAAAAEaNAh0AAAAAANm8vT/v/bd78+kFj+SZsw/LX136rMzx+FYAAACgwynQAQAAAAB0ubuWrc+vX3t7Hly9Oe946dz8xstPy4TenqZjAQAAAIw6BToAAAAAgC41OFjz9/+xJH/+lUWZMXVSPvnW8/PcuTOajgUAAAAwZhToAAAAAAC60PIN2/Kbn7kj3128OheedUyueu3Tc/iUiU3HAgAAABhTCnQAAAAAAF2m1ppf+odbsnTV5lz1356en/upE1JKaToWAAAAwJhToAMAAAAA6DLfXLQiCx/fkD9//TPzumfPbjoOAAAAQGN6mg4AAAAAAMDY+rtvLclxh03OJWcf13QUAAAAgEYp0AEAAAAAdJFbH1qTm5euyVtfeEom9BoRAwAAAN3NdAQAAAAAoIv87beW5PApE3LpeSc0HQUAAACgcQp0AAAAAABd4v7lG/O1hctz2XPnZMrEvqbjAAAAADROgQ4AAAAAoEt84vsPZVJfTy573pymowAAAACMCwp0AAAAAABd4juLV+X8U2bkyKkTm44CAAAAMC4o0AEAAAAAdIEVG7blBys357lzZzQdBQAAAGDcUKADAAAAAOgC/7lkdZLkeQp0AAAAAD+kQAcAAAAA0AW+v2R1pk/uy1nHHdZ0FAAAAIBxQ4EOAAAAAKAL/OcPVuc5Jx+Z3p7SdBQAAACAcUOBDgAAAACgw63ZvCNLV2/JeScf2XQUAAAAgHFFgQ4AAAAAoMNt3LYzSTJj6qSGkwAAAACMLwp0AAAAAAAdbsuOgSTJlIm9DScBAAAAGF8U6AAAAAAAOtzWnbsKdJMV6AAAAAB+jAIdAAAAAECH2/rkHegmKNABAAAA7E6BDgAAAACgwz1ZoDvEHegAAAAAfowCHQAAAABAh1uxcXuSZPrkCQ0nAQAAABhfFOgAAAAAADrcV+59IrOPOCRzZkxpOgoAAADAuKJABwAAAADQwdZt2ZHvPLAqFz/j2JRSmo4DAAAAMK4o0AEAAAAAdLCv3LM8/YM1r3r6cU1HAQAAABh3FOgAAAAAADrYF+96PCceOSVPO/7QpqMAAAAAjDsKdAAAAAAAHWrt5h357mKPbwUAAADYFwU6AAAAAIAOVGvNX3x1UQYGa171jGObjgMAAAAwLinQAQAAAAB0oKu/tzT/9P2Hc8WLTslZxx3WdBwAAACAcamv6QAwFuZceX3TEQBg1IzWn3NLr7p4VM4LAACMvq8vXJ4/+uK9ecWZs/I7F85vOg4AAADAuOUOdAAAAAAAHeTexzbkXZ+6PWced2j+96Vnp7enNB0JAAAAYNxSoAMAAAAA6BDLN2zL5dfckkMnT8hHL/upTJnoISQAAAAAT8X0BAAAAACgA2zZ0Z+3XrMg67fuzD+//bmZdejkpiMBAAAAjHsKdAAAAAAAHeDTtzySux5dn4/84rk567jDmo4DAAAA0BY8whUAAAAAoAM8uGpzDp3cl5efOavpKAAAAABtQ4EOAAAAAKADLFu7NbOPmNJ0DAAAAIC2okAHAAAAANABlq3dktlHHNJ0DAAAAIC2okAHAAAAANDmaq1ZtnZrjlegAwAAABgWBToAAAAAgDa3bsvObNkx4BGuAAAAAMOkQAcAAAAA0MYeXr0l7/rU7UmS02ZNazgNAAAAQHvpazoAAAAAAADD1z8wmI9858H876/dn76envzRJWflBace1XQsAAAAgLaiQAcAAAAA0GbuWrY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      "text/plain": [
       "<Figure size 2520x720 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(35,10))\n",
    "\n",
    "ax1.set_title(\"Histograma de las respuestas de la encuesta de AKAIKE\", {'fontsize': 28})\n",
    "ax1.hist(data, bins=50, density=True) # density=True es para normalizarlo\n",
    "\n",
    "ax2.set_title(\"Función de distribución empírica de las respuestas de la encuesta de AKAIKE\", {'fontsize': 28})\n",
    "ax2.plot(ecdf.x, ecdf.y)\n",
    "\n",
    "fig.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4bd1afa5",
   "metadata": {},
   "source": [
    "$$ $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "877ad567",
   "metadata": {},
   "source": [
    "La empresa reconsiderará el precio si hay evidencia de que las personas estan dispuestas a pagar más de $50.\n",
    "\n",
    "Es decir, estoy buscando ver evidencia de que la media de mi población es mejor que 50.\n",
    "\n",
    "$$ H_0: \\mu \\leq 50 \\quad vs \\quad H_1: \\mu \\gt 50 $$\n",
    "\n",
    "$$ H_0: \\lambda \\geq \\frac{1}{50} \\quad vs \\quad H_1: \\lambda \\lt \\frac{1}{50} $$\n",
    "\n",
    "Siendo la distribución exponencial miembro de la familia exponencial, buscar un estadístico $T$ para $\\lambda$ se puede lograr con el teorema de la factorización.\n",
    "\n",
    "$$ L = \\lambda^n * e^{-\\lambda * \\sum_{i=1}^{n} X_i} $$\n",
    "\n",
    "$$ T = \\sum_{i=1}^{n} X_i $$\n",
    "\n",
    "Mi test de hipótesis se arma como:\n",
    "\n",
    "$$\n",
    "\\delta(\\underline{X}) = \\begin{cases}\n",
    "                          1 \\quad si \\quad \\sum_{i=1}^{100} X_i \\gt k_\\alpha \\\\ \\\\\n",
    "                          0 \\quad si \\quad \\sum_{i=1}^{100} X_i \\leq k_\\alpha\n",
    "                         \\end{cases}\n",
    "$$\n",
    "\n",
    "Calculemos ahora $k_\\alpha$\n",
    "\n",
    "$$ \\lambda \\sum_{i=1}^{n} X_i \\sim \\chi_{2n, \\alpha}^{2} \\Rightarrow  \\lambda \\sum_{i=1}^{100} X_i \\sim \\chi_{200, 0.05}^{2}$$\n",
    "\n",
    "Y juntemos ambos resultados\n",
    "\n",
    "$$\n",
    "\\delta(\\underline{X}) = \\begin{cases}\n",
    "                          1 \\quad si \\quad 2\\lambda\\sum_{i=1}^{100} X_i \\gt \\chi_{200, 0.05}^{2} \\\\ \\\\\n",
    "                          0 \\quad si \\quad 2\\lambda\\sum_{i=1}^{100} X_i \\leq \\chi_{200, 0.05}^{2}\n",
    "                         \\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c081c06c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "¿221.723328 es mayor a 233.99426889232492? False\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import chi2 \n",
    "\n",
    "suma_datos = sum(data)\n",
    "lamb = 1/50\n",
    "izq = 2 * lamb * suma_datos\n",
    "\n",
    "der = chi2.ppf(1 - 0.05, 200)\n",
    "\n",
    "print(f\"¿{izq} es mayor a {der}? {izq > der}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3aede9e9",
   "metadata": {},
   "source": [
    "Por último, el P-Valor lo tenemos que comparar contra $\\alpha = 0.05$ \n",
    "\n",
    "$$\\mathbb{P}\\left(\\chi_{200}^{2} \\gt 221\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4b955a0b",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "¿0.14722538959198506 es mayor a 0.05? True\n"
     ]
    }
   ],
   "source": [
    "p_valor = 1 - chi2.cdf(221, 200)\n",
    "\n",
    "print(f\"¿{p_valor} es mayor a {0.05}? {p_valor > 0.05}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d42522e4",
   "metadata": {},
   "source": [
    "Como el p_valor es mayor al nivel de significación, no puedo recomendar a la empresa subir el precio!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bccf264",
   "metadata": {},
   "source": [
    "$$ $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5724f3d2",
   "metadata": {},
   "source": [
    "Para encontrar el p-valor asintótico quiero hacer uso de la siguiente propipedad\n",
    "\n",
    "> Propiedades Asintóticas de los estimadores de máxima verosimilitud\n",
    "\n",
    "$$ (\\hat{\\lambda}(\\underline{X}) - \\lambda) * \\sqrt{n * \\mathcal{I}(\\lambda)} \\xrightarrow[n\\to\\infty]{\\mathcal{D}} \\mathcal{N}(0,1) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de87d444",
   "metadata": {},
   "source": [
    "Para llegar a eso, primero necesitamos calcular el EMV. Como ya conocemos la función de verosimilitud, solo nos queda...\n",
    "\n",
    "$$ L = \\lambda^n * e^{-\\lambda * \\sum_{i=1}^{n} X_i} $$\n",
    "\n",
    "$$ \\ln(L) = n \\ln(\\lambda) - \\lambda \\sum_{i=1}^{n} X_i $$\n",
    "\n",
    "$$ \\frac{\\partial\\ln(L)}{\\partial\\lambda} = \\frac{n}{\\lambda} - \\sum_{i=1}^{n} X_i $$\n",
    "\n",
    "Igualando a cero, nos queda\n",
    "\n",
    "$$ \\hat{\\lambda} = \\frac{n}{\\sum_{i=1}^{n} X_i} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f5fe12a",
   "metadata": {},
   "source": [
    "Por otro lado, precisamos de la información de Fisher de la distribución exponencial\n",
    "\n",
    "$$ \\mathcal{I}(\\lambda) = - E\\left[ \\frac{\\partial^2}{\\partial^2\\lambda} \\ln(f_X(x)) \\right] $$\n",
    "\n",
    "Vayamos por partes... Primero el $\\ln$, después las derivadas.\n",
    "\n",
    "$$ \\ln(f_X(x)) = \\ln(\\lambda e^{- \\lambda x}) = - \\lambda x \\ln(\\lambda)$$\n",
    "\n",
    "$$ \\frac{\\partial \\ln(f_X(x)))}{\\partial\\lambda} = -x $$\n",
    "\n",
    "$$ \\frac{\\partial^2 \\ln(f_X(x)))}{\\partial^2\\lambda} = -\\frac{x}{\\lambda} $$\n",
    "\n",
    "Ahora sí, solo buscamos la esperanza de $-\\frac{x}{\\lambda}$\n",
    "\n",
    "$$ \\mathcal{I}(\\lambda) = - E\\left[ -\\frac{x}{\\lambda} \\right] = \\frac{1}{\\lambda^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d799ff9",
   "metadata": {},
   "source": [
    "Ahora si, podemos usar la propiedad\n",
    "\n",
    "$$ (\\hat{\\lambda}(\\underline{X}) - \\lambda) * \\sqrt{n * \\mathcal{I}(\\lambda)} \\xrightarrow[n\\to\\infty]{\\mathcal{D}} \\mathcal{N}(0,1) $$\n",
    "\n",
    "$$ \\left(\\frac{n}{\\sum_{i=1}^{n} X_i} - \\lambda\\right) * \\sqrt{n *  \\frac{1}{\\lambda^2}} \\sim \\mathcal{N}(0,1) $$\n",
    "\n",
    "$$ \\delta(\\underline{X}) = \\mathbb{1} \\left\\{ \\left(\\frac{n}{\\sum_{i=1}^{n} X_i} - \\lambda\\right) * \\sqrt{n *  \\frac{1}{\\lambda^2}} \\lt k_\\alpha \\right\\} $$\n",
    "\n",
    "$$ 0.05 = \\lim_{n\\to\\infty} \\mathbb{P}_{\\lambda=\\frac{1}{50}} \\left( \\left(\\frac{n}{\\sum_{i=1}^{n} X_i} - \\lambda\\right) * \\sqrt{n *  \\frac{1}{\\lambda^2}} \\lt Z_{0.05} \\right) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "03faf929",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "¿-0.9797493207390432 es menor a -1.6448536269514729? False\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "\n",
    "suma_datos = sum(data)\n",
    "n = 100\n",
    "lamb = 1/50\n",
    "\n",
    "\n",
    "izq = ((n / suma_datos) - lamb) * (n * (1 / (lamb**2)))**(1/2)\n",
    "der = norm.ppf(0.05)\n",
    "\n",
    "print(f\"¿{izq} es menor a {der}? {izq < der}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47e73e15",
   "metadata": {},
   "source": [
    "Finalmente, el P-Valor lo tenemos que comparar contra $\\alpha = 0.05$ \n",
    "\n",
    "$$\\mathbb{P}\\left( \\left(\\frac{n}{\\sum_{i=1}^{n} X_i} - \\lambda\\right) * \\sqrt{n *  \\frac{1}{\\lambda^2}} \\lt -0.9797 \\right) $$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c7795859",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "¿0.1636049369480293 es mayor a 0.05? True\n"
     ]
    }
   ],
   "source": [
    "p_valor_asintotico = norm.cdf(izq)\n",
    "\n",
    "print(f\"¿{p_valor_asintotico} es mayor a {0.05}? {p_valor_asintotico > 0.05}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "252b65b9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Los valores del p_valor (0.1472) y del p_valor_asintotico (0.1636) se asemejan bastante\n"
     ]
    }
   ],
   "source": [
    "print(f\"Los valores del p_valor ({round(p_valor, 4)}) y del p_valor_asintotico ({round(p_valor_asintotico, 4)}) se asemejan bastante\")"
   ]
  }
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notebook.pdf