08-Independencia_Momentos_Covarianza.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comó se relacionan las marginales, la conjunta y las condicionales?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large P(X_1, X_2) = P(X_1 | X_2=x_2)P(X_2) = P(X_2 | X_1=x_1)P(X_1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "from collections import Counter\n",
    "\n",
    "df = pd.read_csv('alturas-pesos-train.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "tuplas = []\n",
    "for line in df[['Peso', 'Altura']].values:\n",
    "    tuplas.append(tuple(line))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "frecuencia_conjunta = dict(Counter(tuplas))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{(60, 160): 453, (95, 180): 201, (75, 170): 394, (55, 160): 272, (55, 150): 59, (80, 175): 351, (70, 170): 279, (55, 155): 302, (85, 180): 239, (60, 165): 235, (80, 165): 98, (95, 185): 153, (75, 165): 209, (65, 170): 128, (65, 160): 305, (40, 150): 16, (40, 145): 22, (70, 160): 108, (65, 165): 402, (85, 170): 176, (85, 165): 23, (50, 155): 208, (90, 175): 262, (30, 145): 1, (45, 150): 93, (60, 155): 179, (90, 170): 36, (90, 180): 335, (110, 195): 11, (85, 185): 30, (65, 155): 55, (85, 175): 408, (70, 165): 337, (60, 170): 31, (60, 150): 16, (90, 190): 5, (100, 185): 97, (45, 155): 46, (50, 150): 133, (105, 185): 18, (80, 170): 369, (35, 145): 3, (95, 175): 58, (50, 160): 60, (75, 175): 134, (100, 190): 48, (75, 160): 42, (55, 165): 69, (95, 190): 38, (70, 175): 44, (55, 145): 4, (90, 185): 119, (105, 195): 13, (50, 145): 16, (100, 175): 9, (80, 180): 63, (100, 180): 33, (45, 145): 33, (110, 190): 8, (115, 200): 2, (45, 160): 6, (105, 190): 27, (65, 175): 8, (70, 155): 8, (40, 140): 4, (30, 140): 1, (100, 195): 2, (115, 195): 3, (75, 180): 6, (40, 160): 1, (65, 150): 5, (55, 170): 6, (80, 185): 4, (80, 160): 2, (50, 165): 4, (40, 155): 2, (35, 140): 3, (85, 160): 1, (105, 200): 1, (45, 140): 2, (35, 155): 1, (120, 200): 1, (105, 180): 3, (95, 195): 2, (110, 200): 1, (115, 190): 3, (70, 180): 1, (75, 155): 1}\n"
     ]
    }
   ],
   "source": [
    "print(frecuencia_conjunta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejemplo\n",
    "\n",
    "$\\large P(Peso= 80, Altura =165) = P(Peso=80|Altura=165)P(Altura=165)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Izquierda del igual $P(Peso= 80, Altura =165)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.01225\n"
     ]
    }
   ],
   "source": [
    "# Si lo obtenemos de la conjunta directamente\n",
    "P_80_165 = frecuencia_conjunta[(80, 165)]/len(df)\n",
    "print(P_80_165)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derecha del igual $P(Peso=80|Altura=165)P(Altura=165)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Cant_alt_165 = (df['Altura'] == 165).sum()\n",
    "Cant_peso_80_alt_165 = (df[df['Altura'] == 165]['Peso'] == 80).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.07116920842411038\n"
     ]
    }
   ],
   "source": [
    "# Primer parte del producto \n",
    "P_80_dado_165 = Cant_peso_80_alt_165/Cant_alt_165\n",
    "print(P_80_dado_165)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.172125"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Segunda parte del producto \n",
    "P_alt_165 = Cant_alt_165/len(df)\n",
    "P_alt_165"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01225"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Resultado del producto que coincide como es de esperar con el \n",
    "P_alt_165*P_80_dado_165"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Notar como se hizo el cálculo\n",
    "\n",
    "#### Cant_alt_165/len(df) * Cant_peso_80_alt_165/Cant_alt_165 = Cant_peso_80_alt_165/len(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Independencia"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P(X_1, X_2) = P(X_1 | X_2=x_2)P(X_2)$\n",
    "\n",
    "Si $X_1$ no depende de $X_2$ $P(X_1 | X_2=x_2)$ = $P(X_1)$\n",
    "\n",
    "$\\large P(X_1, X_2) = P(X_1)P(X_2)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# De conjunta a marginal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large P(X_1) = \\sum_{x_2}{P(X_1, X_2)}$\n",
    "\n",
    "$\\large P(Pesos) = \\sum_{Alturas}{P(Pesos, Alturas)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from helper import plot_joint_hists_dicts\n",
    "_ = plot_joint_hists_dicts(frecuencia_conjunta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probabilidad Total\n",
    "$\\large P(X_1) = \\sum_{x_2}{P(X_1| X_2)P(X_2)}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large P(Pesos = p) = P(Pesos = p| Mujer)P(Mujer) + P(Pesos| Hombre)P(Hombre)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A $𝑃(𝑋1|𝑋2)$ se lo llama likelihood"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.112875"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 𝑃(𝑃𝑒𝑠𝑜𝑠=𝑝)\n",
    "(df['Peso'] == 65).sum()/len(df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11287499999999999"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 𝑃(𝑃𝑒𝑠𝑜𝑠=𝑝|𝑀𝑢𝑗𝑒𝑟)𝑃(𝑀𝑢𝑗𝑒𝑟)+𝑃(𝑃𝑒𝑠𝑜𝑠|𝐻𝑜𝑚𝑏𝑟𝑒)𝑃(𝐻𝑜𝑚𝑏𝑟𝑒)\n",
    "((df['Peso'] == 65) & (df['Genero'] == 'Mujer')).sum()/(df['Genero']=='Mujer').sum() * (df['Genero']=='Mujer').sum()/len(df) + \\\n",
    "((df['Peso'] == 65) & (df['Genero'] == 'Hombre')).sum()/(df['Genero']=='Hombre').sum() * (df['Genero']=='Hombre').sum()/len(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vector de medias"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$E\\{[X_1, ..., X_n]\\} = [\\mu_{x_1}, ..., \\mu_{x_n}]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(168.10875, 72.814375)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['Altura'].mean(), df['Peso'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Covarianza de 2 Variables aleatorias"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$cov(X_1, X_2) = E[(X_1 - \\mu_{x_1})((X_2 - \\mu_{x_2})] = E\\{X_1X_2\\} - \\mu_{X_1}\\mu_{X_2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Que pasa si $X_1$ es igual a $X_2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$cov(X_1, X_2) = \\sigma_{x_1}^2 = \\sigma_{x_2}^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$cov(X_1, X_2) = \\sum_{(x_1, x_2) \\in S} P(X_1=x_1, X_2=x_2)(x_1-\\mu_{x_1})(x_2- \\mu_{x_2})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cual es minimo positivo?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "168.10875 72.814375\n"
     ]
    }
   ],
   "source": [
    "conjunta_prob = {al:c/len(df) for al, c in frecuencia_conjunta.items()}\n",
    "mu_alturas = df['Altura'].mean()\n",
    "mu_pesos = df['Peso'].mean()\n",
    "print(mu_alturas, mu_pesos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "133.01956171875\n"
     ]
    }
   ],
   "source": [
    "cov = 0\n",
    "for (pe, al), p in conjunta_prob.items():\n",
    "    cov = cov + (pe-mu_pesos)*(al-mu_alturas)*p\n",
    "print(cov)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Es alta o baja?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coeficiente de correlación\n",
    "Mide la relación **lineal** entre dos V.As"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Si normalizo obtengo el coeficiente de correlación"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\large \\rho(X_1, X_2) = \\frac{cov(X_1, X_2)}{\\sigma_{X_1} \\sigma_{X_2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "std_alturas = df['Altura'].std()\n",
    "std_pesos = df['Peso'].std()\n",
    "corr_cof = cov/(std_alturas*std_pesos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9105196990184911"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corr_cof"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Muy alta!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Diferencia de correlación con entropía cruzada o divergencia KL?\n",
    "\n",
    "- La entropía cruzada trabaja sobre el mismo espacio muestral mientras que la correlación no"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matriz de covarianza en vectores aleatorios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\vec{X} = [X_1, X_2, X_3, ..., X_N]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\Sigma_{\\vec{X}} = \n",
    "\\begin{bmatrix}\n",
    "    cov(x_1,x_1) & cov(x_1,x_2) & cov(x_1,x_3) & \\dots  & cov(x_1,x_N) \\\\\n",
    "    cov(x_2,x_1) & cov(x_2,x_2) & cov(x_2,x_3) & \\dots  & cov(x_2,x_N) \\\\\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    cov(x_N,x_1) & cov(x_N,x_2) & cov(x_N,x_3) & \\dots  & cov(x_N,x_N)\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\Sigma_{\\vec{X}} = \n",
    "\\begin{bmatrix}\n",
    "    \\sigma_{x_1}^2 & cov(x_1,x_2) & cov(x_1,x_3) & \\dots  & cov(x_1,x_N) \\\\\n",
    "    cov(x_2,x_1) & \\sigma_{x_2}^2 & cov(x_2,x_3) & \\dots  & cov(x_2,x_N) \\\\\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    cov(x_N,x_1) & cov(x_N,x_2) & cov(x_N,x_3) & \\dots  & \\sigma_{x_N}^2\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Que pasa cuando todas las V.A son independientes?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\Sigma_{\\vec{X}} = \n",
    "\\begin{bmatrix}\n",
    "    \\sigma_{x_1}^2 & 0 & 0 & \\dots  & 0 \\\\\n",
    "    0 & \\sigma_{x_2}^2 & 0 & \\dots  & 0 \\\\\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    0 & 0 & 0 & \\dots  & \\sigma_{x_N}^2\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matriz de correlación"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\Sigma_{\\vec{X}} = \n",
    "\\begin{bmatrix}\n",
    "    1 & \\rho_{x_1,x_2} & \\rho_{x_1,x_3} & \\dots  & \\rho_{x_1,x_N} \\\\\n",
    "    \\rho_{x_2,x_1} & 1 & \\rho_{x_2,x_3} & \\dots  & \\rho_{x_2,x_N} \\\\\n",
    "    \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "    cov{x_N,x_1} & \\rho_{x_N,x_2} & \\rho_{x_N,x_3} & \\dots  & 1\n",
    "\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Peso</th>\n",
       "      <th>Altura</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Peso</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.910634</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Altura</th>\n",
       "      <td>0.910634</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            Peso    Altura\n",
       "Peso    1.000000  0.910634\n",
       "Altura  0.910634  1.000000"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.corr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2 = df.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "df2['Genero'] = df['Genero'] == 'Hombre'\n",
    "# Cambiar Hombre por Mujer y ver que pasa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Genero</th>\n",
       "      <th>Peso</th>\n",
       "      <th>Altura</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Genero</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.793309</td>\n",
       "      <td>0.684495</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Peso</th>\n",
       "      <td>0.793309</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.910634</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Altura</th>\n",
       "      <td>0.684495</td>\n",
       "      <td>0.910634</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          Genero      Peso    Altura\n",
       "Genero  1.000000  0.793309  0.684495\n",
       "Peso    0.793309  1.000000  0.910634\n",
       "Altura  0.684495  0.910634  1.000000"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.corr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# En numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>7978</th>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <th>7979</th>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <th>7980</th>\n",
       "      <td>True</td>\n",
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       "      <td>190</td>\n",
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       "    <tr>\n",
       "      <th>7986</th>\n",
       "      <td>False</td>\n",
       "      <td>80</td>\n",
       "      <td>175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7987</th>\n",
       "      <td>True</td>\n",
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       "    <tr>\n",
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       "      <td>170</td>\n",
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       "      <th>7990</th>\n",
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       "      <td>155</td>\n",
       "    </tr>\n",
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       "      <th>7991</th>\n",
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       "      <td>False</td>\n",
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       "      <td>True</td>\n",
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       "      <td>180</td>\n",
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       "    <tr>\n",
       "      <th>7997</th>\n",
       "      <td>True</td>\n",
       "      <td>75</td>\n",
       "      <td>170</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7998</th>\n",
       "      <td>False</td>\n",
       "      <td>65</td>\n",
       "      <td>165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7999</th>\n",
       "      <td>True</td>\n",
       "      <td>80</td>\n",
       "      <td>170</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>8000 rows × 3 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "      Genero  Peso  Altura\n",
       "0      False    60     160\n",
       "1       True    95     180\n",
       "2       True    75     170\n",
       "3      False    55     160\n",
       "4      False    55     150\n",
       "5      False    60     160\n",
       "6       True    80     175\n",
       "7      False    70     170\n",
       "8      False    55     155\n",
       "9       True    85     180\n",
       "10     False    60     165\n",
       "11      True    80     165\n",
       "12      True    95     185\n",
       "13     False    75     165\n",
       "14     False    65     170\n",
       "15      True    85     180\n",
       "16     False    65     160\n",
       "17     False    40     150\n",
       "18     False    40     145\n",
       "19     False    70     160\n",
       "20     False    60     160\n",
       "21     False    60     160\n",
       "22     False    65     165\n",
       "23      True    85     170\n",
       "24      True    75     165\n",
       "25      True    95     180\n",
       "26      True    85     165\n",
       "27     False    55     155\n",
       "28     False    55     155\n",
       "29     False    50     155\n",
       "...      ...   ...     ...\n",
       "7970   False    65     160\n",
       "7971   False    55     155\n",
       "7972   False    60     155\n",
       "7973    True    90     175\n",
       "7974    True    70     165\n",
       "7975    True    75     160\n",
       "7976    True    75     160\n",
       "7977   False    65     155\n",
       "7978    True    80     175\n",
       "7979   False    40     145\n",
       "7980    True   105     190\n",
       "7981    True   100     185\n",
       "7982    True    70     160\n",
       "7983    True    80     175\n",
       "7984   False    70     165\n",
       "7985   False    60     155\n",
       "7986   False    80     175\n",
       "7987    True    85     170\n",
       "7988   False    65     170\n",
       "7989    True    80     170\n",
       "7990   False    55     155\n",
       "7991    True    85     180\n",
       "7992    True    90     180\n",
       "7993    True   100     185\n",
       "7994    True    90     175\n",
       "7995   False    60     160\n",
       "7996    True    85     180\n",
       "7997    True    75     170\n",
       "7998   False    65     165\n",
       "7999    True    80     170\n",
       "\n",
       "[8000 rows x 3 columns]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'df2' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-19-2068723f1203>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mdf2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'df2' is not defined"
     ]
    }
   ],
   "source": [
    "df2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.        , 0.7933089 , 0.68449547],\n",
       "       [0.7933089 , 1.        , 0.91063353],\n",
       "       [0.68449547, 0.91063353, 1.        ]])"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# En numpy\n",
    "np.corrcoef(np.array(df2.values.T, dtype=float))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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