Practica 7 (nueva) - Ejercicio 7.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T01:17:01.656953Z",
     "start_time": "2019-10-29T01:17:01.642Z"
    }
   },
   "source": [
    "![](./img/practica7/7.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**a )**\n",
    "$$\\large X \\sim Ge(\\theta)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**b )**\n",
    "\n",
    ">|Estimador de **Máxima Verosimilitud**|\n",
    ">|---|\n",
    ">|$$\\huge \\hat \\theta_{MV} = \\frac {1}{1 - \\sum_{i=1}^n X_i}$$|\n",
    "\n",
    "\n",
    ">|Estimador de **Momentos**|\n",
    ">|---|\n",
    ">|$$\\huge \\hat \\theta_{m} = \\frac {n}{\\sum_{i=1}^n X_i}$$|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**c )**\n",
    "\n",
    "$$P_{\\hat \\theta_{MV}} (X \\geq 5) = 1 - P_{\\hat \\theta_{MV}} (X \\leq 4)$$\n",
    "\n",
    "CMF de Geométrica: $P(X \\leq k) = 1 - (1-p)^k$\n",
    "\n",
    ">|Estimador de $P(X \\geq 5)$ con $\\hat \\theta_{MV}$|\n",
    ">|---|\n",
    ">|$$\\huge P_{\\hat \\theta_{MV}} (X \\geq 5) = \\left(1 - \\hat \\theta_{MV} \\right)^4 $$|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**d )** Sin asumir que es geométrica la pensé a partir de 4 eventos individuales:\n",
    "\n",
    "$$P(X\\leq4) = P(X = 1) + P(X = 2) +P(X = 3) + P(X = 4)$$\n",
    "\n",
    "> Donde\n",
    ">\n",
    "> $P(X = 1) = \\theta$\n",
    ">\n",
    "> $P(X = 2) = \\theta (1-\\theta)$\n",
    ">\n",
    "> $P(X = 3) = \\theta (1-\\theta)^2$\n",
    ">\n",
    "> $P(X = 4) = \\theta (1-\\theta)^3$\n",
    "\n",
    "Que resulta ser el mismo polinomio que usando la función acumulada de la geométrica:\n",
    "\n",
    "$$(1-\\theta)^4 = \\theta + \\theta (1-\\theta) + \\theta (1-\\theta)^2 +\\theta (1-\\theta)^3$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Evaluación con datos:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:35:27.719903Z",
     "start_time": "2019-10-29T02:35:27.663Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>5</li>\n",
       "\t<li>12</li>\n",
       "\t<li>1</li>\n",
       "\t<li>7</li>\n",
       "\t<li>6</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 5\n",
       "\\item 12\n",
       "\\item 1\n",
       "\\item 7\n",
       "\\item 6\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 5\n",
       "2. 12\n",
       "3. 1\n",
       "4. 7\n",
       "5. 6\n",
       "6. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  5 12  1  7  6  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta <- 0.1\n",
    "datos <- rgeom(200000, prob=theta)\n",
    "head(datos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:35:28.336165Z",
     "start_time": "2019-10-29T02:35:28.235Z"
    }
   },
   "outputs": [
    {
     "data": {
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y67rLCwcNeuXXUwHAAA1Vd12P3P//xPUVHR2rVrp06deuqpp/7xj38cMmRI27Zt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\n7cWekZQ6fQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title “Histogram of datos”"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(datos)\n",
    "abline(v=mean(datos), col='red')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:50:10.130410Z",
     "start_time": "2019-10-29T02:50:10.110Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0,111097285671116"
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      "text/latex": [
       "0,111097285671116"
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      "text/plain": [
       "[1] 0,1110973"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "theta_hat_MV <- n / sum(datos)\n",
    "theta_hat_MV"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:50:10.260222Z",
     "start_time": "2019-10-29T02:50:10.239Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0,111097285671116"
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      "text/latex": [
       "0,111097285671116"
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       "[1] 0,1110973"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n <- length(datos)\n",
    "theta_hat_m <- n / sum(datos)\n",
    "theta_hat_m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:50:25.310896Z",
     "start_time": "2019-10-29T02:50:25.292Z"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "0,624333918069707"
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       "[1] 0,6243339"
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     "metadata": {},
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   ],
   "source": [
    "# P(X >= 5) = \n",
    "invAcumuladaConModelo <- function(p){\n",
    "    return ((1 - p)^4)\n",
    "}\n",
    "invAcumuladaConModelo(theta_hat_MV)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T02:50:23.481974Z",
     "start_time": "2019-10-29T02:50:23.454Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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      "text/latex": [
       "0,624333918069707"
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       "[1] 0,6243339"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# P(X >= 5) = \n",
    "invAcumuladaSinModelo <- function(p) {\n",
    "    # = 1 - ( P(X=1) + P(X=2) + P(X=3) + P(X=4) )\n",
    "    return (1 - (p + p*(1-p) + p*(1-p)^2 + p*(1-p)^3))\n",
    "}\n",
    "invAcumuladaSinModelo(theta_hat_MV)"
   ]
  },
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