Aula17_Atividade_CombinacaoDeVariaveisAleatorias.ipynb

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   "source": [
    "___\n",
    "# Atividade: <p> <center> Propriedades de esperança e variância <p> <center> para Combinação linear de variáveis aleatórias\n",
    "___\n",
    "\n",
    "## Aula 17\n",
    "\n",
    "**Objetivo da aula:** Ao final desta aula, o aluno deve ser capaz de:\n",
    " * Reconhecer variáveis aleatórias de interesse em um problema qualquer; e\n",
    " * Aplicar propriedades de esperança e variância quando se tem combinação linear entre variáveis aleatórias\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"indice\"></div>\n",
    "\n",
    "## Índice\n",
    "\n",
    "- [Motivação teórica: estudo das f.d.p.'s](#motivacao)\n",
    "    - [Exercício 1](#ex1)\n",
    "\n",
    "<p> \n",
    "    \n",
    "- [PROBLEMA: O que significa fazer $X+Y$?](#significa)\n",
    "\n",
    "<p> \n",
    "    \n",
    "- [SIMULAÇÃO: Programando valores para representar $X$, $Y$ e $X+Y$](#simulacao)\n",
    "    - [Simulação com uso de *loop*](#simulacao1)\n",
    "        - [Exercício 2 - Média e Variância via `numpy`](#ex2)\n",
    "    - [Simulação sem uso de *loop* ](#simulacao2)\n",
    "        - [Exercício 3 - Média e Variância via `pandas`](#ex3)\n",
    "\n",
    "<p> \n",
    "\n",
    "- [TEORIA: Contrastando respostas utilizando as propriedades](#teoria)       \n",
    "    - [Propriedades de esperança e variância em combinação linear de variáveis aleatórias](#propriedades)       \n",
    "        - [Exercício 4 - Média e Variância via propriedades](#ex4)\n",
    "\n",
    "<p>\n",
    "    \n",
    "- [Lista de comandos utilizados neste notebook](#comandos)\n",
    "___"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"motivacao\"></div>\n",
    "\n",
    "# Motivação teórica: estudo das f.d.p.'s\n",
    "\n",
    "A seguir, considere duas variáveis aleatórias $X$ e $Y$, ambas contínuas, cujo objetivo seja somá-las, ou seja, obter $X+Y$!\n",
    "\n",
    "Vamos criar um experimento considerando as seguintes distribuições e parâmetros para $X$ e $Y$, respectivamente:\n",
    "\n",
    " $X \\sim$ Uniforme$(4,7)$, ou seja, $a=4$ e $b=7$\n",
    "\n",
    "e \n",
    "\n",
    " $Y \\sim$ Exponencial$(1/3)$, ou seja, $\\lambda=1/3$ e $\\mu=3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div id=\"ex1\"></div>\n",
    "\n",
    "### EXERCÍCIO 1\n",
    "\n",
    "Inicialmente, construa um gráfico de ambas as distribuições univariadamente. Veja **Aula 12** caso tenha dúvidas com essas duas distribuições."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1000x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ESCREVA SEU CÓDIGO AQUI\n",
    "a = 4\n",
    "b = 7\n",
    "mu = 3\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.subplot(121)\n",
    "plt.plot(np.arange(a, b, 0.01), stats.uniform.pdf(np.arange(a, b, 0.01), loc=a, scale = b-a))\n",
    "plt.subplot(122)\n",
    "plt.plot(np.arange(a, b, 0.01), stats.expon.pdf(np.arange(a, b, 0.01), scale = mu))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"significa\"></div>\n",
    "\n",
    "# PROBLEMA: O que significa fazer $X + Y$ ?\n",
    "\n",
    "Considere, **por exemplo**, que um produto passe por duas etapas na linha de produção e o tempo em cada etapa seja mensurados. Assim, assuma que:\n",
    "\n",
    "  * $X$ represente o tempo, em minutos, que um produto demora para passar pela etapa $1$; e\n",
    "  * $Y$ represente o tempo, em minutos, que o mesmo produto demora para passar pela etapa $2$ na linha de produção.\n",
    "  \n",
    "Assim, $W=X+Y$ deverá representar o tempo total que um produto demora para ficar pronto se esse precisa apenas dessas duas etapas na linha de produção.\n",
    "\n",
    "O que vamos entender agora é como podemos analisar o **valor esperado** e a **variância** de $W$ a partir de dados e depois considerando propriedades de esperança e variância."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"simulacao\"></div>\n",
    "\n",
    "# SIMULAÇÃO: Programando valores para representar $X$, $Y$ e $X+Y$\n",
    "\n",
    "Para estudarmos como ficaria o valor esperado e a variância de $W=X+Y$ a partir de dados, devemos simular alguns valores hipotéticos para representar $X$ conforme a uniforme e $Y$ conforme a exponencial mencionada acima. \n",
    "\n",
    "Para tanto, no `scipy.stats`, a função que gera uma amostra de valores conforme uma distribuição qualquer é a função `rvs()` (de *random variable sample*). No nosso exemplo, vamos utilizar `stats.uniform.rvs()` ou `stats.expon.rvs` com os valores dos parâmetros adequadamente aos citados anteriormente.\n",
    "\n",
    "Assim, faremos:\n",
    "  * Simular $n$ valores para construir uma amostra de $X$ utilizando a distribuição uniforme\n",
    "  * Simular $n$ valores para construir uma amostra de $Y$ utilizando a distribuição exponencial\n",
    "  * A cada linha, somar os valores de $x$ e de $y$ para construir a coluna $w$, já que $W = X + Y$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div id=\"simulacao1\"></div>\n",
    "\n",
    "## Simulação com uso de *loop*:\n",
    "\n",
    "Simulação mais \"lenta\" por fazer uso de `for`. Aqui, será simulado um valor por vez para $x$ e para $y$, calculado a soma e aguardado em $w$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 50000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_all = []\n",
    "y_all = []\n",
    "w_all = []\n",
    "\n",
    "a = 4  #Início de valor da Uniforme\n",
    "b = 7  #Final de valor da Uniforme\n",
    "mu = 3 #Média da Exponencial\n",
    "\n",
    "for i in range(n):\n",
    "    x = stats.uniform.rvs(loc=a, scale=b-a) #rvs: random variable sample (gera valor aleatório a partir do modelo uniforme)\n",
    "    y = stats.expon.rvs(scale=mu) #rvs: random variable sample (gera valor aleatório a partir do modelo exponencial)\n",
    "    w = x+y\n",
    "    \n",
    "    # Guarda os n valores em lista\n",
    "    x_all.append(x)\n",
    "    y_all.append(y)\n",
    "    w_all.append(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dependendo das distribuições somadas, a nova distribuição resultante pode apresentar um outro formato. Ou seja, somar uma uniforme e uma exponencial, pode resultar numa terceira distribuição teórica que pode ou não ser facilmente conhecida na literatura estatística. \n",
    "\n",
    "Vamos visualizar o histograma dos valores de $x$, de $y$ e de $w$, esse último que deve indicar o formato da distribuição resultante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1800x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(18, 5))\n",
    "\n",
    "plt.subplot(131)\n",
    "plt.hist(x_all, bins=50, density=True,\n",
    "         color='blue', edgecolor='white', \n",
    "         linewidth=1, alpha=0.8)\n",
    "plt.ylabel('densidade')\n",
    "plt.xlabel('valores simulados para x')\n",
    "plt.title('Simulação - Uniforme')\n",
    "#plt.ylim(0,0.35)\n",
    "\n",
    "plt.subplot(132)\n",
    "plt.hist(y_all, bins=50, density=True,\n",
    "         color='orange', edgecolor='white', \n",
    "         linewidth=1, alpha=0.8)\n",
    "plt.ylabel('densidade')\n",
    "plt.xlabel('valores simulados para y')\n",
    "plt.title('Simulação - Exponencial')\n",
    "plt.ylim(0,0.35)\n",
    "\n",
    "plt.subplot(133)\n",
    "plt.hist(w_all, bins=50, density=True,\n",
    "         color='red', edgecolor='white', \n",
    "         linewidth=1, alpha=0.8)\n",
    "plt.ylabel('densidade')\n",
    "plt.xlabel('valores calculados para w')\n",
    "plt.title('W=X+Y')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**Pontos importantes:** \n",
    "\n",
    "Implementar $X+Y$ é somar os valores de $X$ com os valores de $Y$, linha a linha, e obter uma nova distribuição, cuja área também valerá $1$!!!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div id=\"ex2\"></div>\n",
    "\n",
    "### EXERCÍCIO 2 - Média e Variância via `numpy`\n",
    "\n",
    "Considerando os valores de `x_all`, `y_all` e `w_all`, calcule a média e variância de cada variável. Ainda, calcule covariância e correlação entre `x_all` e `y_all`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Média de X:  5.496632376720039\n",
      "Média de Y:  3.0075747073341597\n",
      "Média de W:  8.5042070840542\n",
      "Variância de X:  0.7481141902540323\n",
      "Variância de Y:  9.053956025460028\n",
      "Variância de W:  9.796807597377331\n",
      "Covariância entre X e Y:  [[ 7.48129153e-01 -2.63136180e-03]\n",
      " [-2.63136180e-03  9.05413711e+00]]\n",
      "Correlação entre X e Y:  [[ 1.         -0.00101104]\n",
      " [-0.00101104  1.        ]]\n"
     ]
    }
   ],
   "source": [
    "# ESCREVA SEU CÓDIGO AQUI\n",
    "x_avg = np.mean(x_all)\n",
    "y_avg = np.mean(y_all)\n",
    "w_avg = np.mean(w_all)\n",
    "x_var = np.var(x_all)\n",
    "y_var = np.var(y_all)\n",
    "w_var = np.var(w_all)\n",
    "xy_cov = np.cov(x_all, y_all)\n",
    "xy_corr = np.corrcoef(x_all, y_all)\n",
    "\n",
    "print('Média de X: ', x_avg)\n",
    "print('Média de Y: ', y_avg)\n",
    "print('Média de W: ', w_avg)\n",
    "print('Variância de X: ', x_var)\n",
    "print('Variância de Y: ', y_var)\n",
    "print('Variância de W: ', w_var)\n",
    "print('Covariância entre X e Y: ', xy_cov)\n",
    "print('Correlação entre X e Y: ', xy_corr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "\n",
    "<div id=\"simulacao2\"></div>\n",
    "\n",
    "## Simulação sem uso de *loop*:\n",
    "\n",
    "Agora, ao invés de fazer um *loop* para gerar $n$ amostras em Python, podemos pedir à função `rvs()` que realize $n$ sorteios diretamente utilizando o **argumento** `size = n`. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observe que os resultados gerados aqui não se alteram, de maneira geral, mas o desempenho é melhor (mais rápido)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_All = stats.uniform.rvs(loc=a, scale=b-a, size=n)\n",
    "y_All = stats.expon.rvs(scale=mu, size=n)\n",
    "\n",
    "# note que estamos somando elemento a elemento de x_All e y_All e obtendo um novo array\n",
    "w_All = x_All + y_All "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div id=\"ex3\"></div>\n",
    "\n",
    "### EXERCÍCIO 3 - Média e Variância via `pandas`\n",
    "\n",
    "Os valores de `x_All`, `y_All` e `w_All`, a seguir, serão colocados em um *dataframe* chamando `df`.\n",
    "\n",
    "Considerando as variáveis `X`, `Y` e `W` desse *dataframe*, calcule a média e variância de cada variável. Ainda, calcule covariância e correlação entre `X` e `Y`.\n",
    "\n",
    "Faça todas as contas utilizando comando do *pandas*.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {
    "scrolled": true
   },
   "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>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>W</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4.636816</td>\n",
       "      <td>3.102850</td>\n",
       "      <td>7.739666</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.512141</td>\n",
       "      <td>1.558035</td>\n",
       "      <td>6.070175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.654108</td>\n",
       "      <td>3.941518</td>\n",
       "      <td>8.595627</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>5.372538</td>\n",
       "      <td>2.375290</td>\n",
       "      <td>7.747828</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>6.674419</td>\n",
       "      <td>12.203624</td>\n",
       "      <td>18.878043</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          X          Y          W\n",
       "0  4.636816   3.102850   7.739666\n",
       "1  4.512141   1.558035   6.070175\n",
       "2  4.654108   3.941518   8.595627\n",
       "3  5.372538   2.375290   7.747828\n",
       "4  6.674419  12.203624  18.878043"
      ]
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.DataFrame()\n",
    "df['X'] = x_All\n",
    "df['Y'] = y_All\n",
    "df['W'] = w_All\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Média de X:  5.498172759235396\n",
      "Média de Y:  2.994122838350454\n",
      "Média de W:  8.492295597585848\n",
      "Variância de X:  0.7506035121370223\n",
      "Variância de Y:  8.997245963813336\n",
      "Variância de W:  9.772278272164046\n",
      "Covariância entre X e Y:  0.012214398106852981\n",
      "Correlação entre X e Y:  0.004700152814568416\n"
     ]
    }
   ],
   "source": [
    "# ESCREVA SEU CÓDIGO AQUI\n",
    "x_avg = df['X'].mean()\n",
    "y_avg = df['Y'].mean()\n",
    "w_avg = df['W'].mean()\n",
    "x_var = df['X'].var()\n",
    "y_var = df['Y'].var()\n",
    "w_var = df['W'].var()\n",
    "xy_cov = df['X'].cov(df['Y'])\n",
    "xy_corr = df['X'].corr(df['Y'])\n",
    "\n",
    "print('Média de X: ', x_avg)\n",
    "print('Média de Y: ', y_avg)\n",
    "print('Média de W: ', w_avg)\n",
    "print('Variância de X: ', x_var)\n",
    "print('Variância de Y: ', y_var)\n",
    "print('Variância de W: ', w_var)\n",
    "print('Covariância entre X e Y: ', xy_cov)\n",
    "print('Correlação entre X e Y: ', xy_corr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"teoria\"></div>\n",
    "\n",
    "\n",
    "# TEORIA: Contrastando respostas utilizando as propriedades\n",
    "\n",
    "Dica: Reveja as aulas de **modelos probabilísticos contínuos** - uniforme e exponencial\n",
    "\n",
    "Relembrando:\n",
    "\n",
    "### Para a Uniforme:\n",
    "\n",
    "Utilizando o resultado teórico da distribuição Uniforme, temos que o valor esperado é dado por:\n",
    "\n",
    "$$E(X) = \\frac{a + b}{2} = 5.5$$\n",
    "\n",
    "Já a variância de uma variável com distribuição Uniforme é dada por:\n",
    "\n",
    "$$Var(X) = \\frac{(b-a)^2}{12} = 0.75$$\n",
    "\n",
    "\n",
    "### Para a exponencial:\n",
    "\n",
    "Utilizando o resultado teórica da distribuição Exponencial, temos que o valor esperado é dado por:\n",
    "\n",
    "$$E(Y) = \\mu = 3$$\n",
    "\n",
    "Já a variância de uma variável com distribuição Exponencial é dada por:\n",
    "\n",
    "$$Var(Y) = \\mu^2 = 9$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"propriedades\"></div>\n",
    "\n",
    "## Propriedades de esperança e variância em combinação linear de variáveis aleatórias\n",
    "\n",
    "\n",
    "### E(W): Valor esperado da W calculada a partir da teoria\n",
    "\n",
    "**Previsão teórica:**\n",
    "\n",
    "**$\\qquad E(W) = E(X)  + E(Y)$**\n",
    "\n",
    "$\\quad$\n",
    "\n",
    "### Var(W): Variância da W calculada a partir da teoria\n",
    "\n",
    "**Previsão teórica:**\n",
    "\n",
    "**$\\qquad Var(W) = Var(X) + Var(Y) + 2 * Cov(X,Y)$**\n",
    "\n",
    "$\\quad$\n",
    "\n",
    "Assuma que as variáveis $X$ e $Y$ são **INDEPENDENTES** até mesmo pela maneira que foram geradas.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div id=\"ex4\"></div>\n",
    "\n",
    "### EXERCÍCIO 4 - Média e Variância via propriedades\n",
    "\n",
    "Considerando os resultados teóricos para $E(X)$, $E(Y)$, $Var(X)$, $Var(Y)$ e $Cov(X,Y)$, calcule o valor esperado e a variância de $W$. \n",
    "\n",
    "Compare os valores obtidos para $E(W)$ e $Var(W)$ com os obtidos nos exercícios [Exercício 2](#ex2) e [Exercício 3](#ex3).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.49229559758585"
      ]
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ESCREVA SEU CÓDIGO AQUI\n",
    "e_w = x_avg + y_avg\n",
    "e_w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.772278272164064"
      ]
     },
     "execution_count": 219,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var_w = x_var + y_var + 2*xy_cov\n",
    "var_w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparando os resultados SIMULADOS com os calculados por PROPRIEDADES:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Média de W dado:  8.492295597585848\n",
      "Média de W prop:  8.49229559758585\n",
      "Variância de W:  9.772278272164046\n",
      "Variância de W prop:  9.772278272164064\n"
     ]
    }
   ],
   "source": [
    "print('Média de W dado: ', w_avg)\n",
    "print('Média de W prop: ', e_w)\n",
    "print('Variância de W: ', w_var)\n",
    "print('Variância de W prop: ', var_w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "<div id=\"comandos\"></div>\n",
    "\n",
    "## Lista de comandos utilizados neste notebook\n",
    "\n",
    "Coloque aqui a lista de novos comandos deste notebook:\n",
    "\n",
    " * \n",
    " "
   ]
  }
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