{"id":26870,"date":"2024-05-25T22:48:04","date_gmt":"2024-05-25T22:48:04","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26870"},"modified":"2024-05-25T22:49:15","modified_gmt":"2024-05-25T22:49:15","slug":"distribuicoes-continuas-de-probabilidade","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/distribuicoes-continuas-de-probabilidade\/","title":{"rendered":"Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Distribui\u00e7\u00f5es Cont\u00ednuas de Probabilidade<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Resumo<\/strong><br \/><em>Aqui examinaremos em profundidade o conceito de distribui\u00e7\u00f5es cont\u00ednuas de probabilidade, destacando as caracter\u00edsticas e usos das cinco mais conhecidas: a distribui\u00e7\u00e3o exponencial, a distribui\u00e7\u00e3o uniforme retangular, a distribui\u00e7\u00e3o normal (Gaussiana), a distribui\u00e7\u00e3o Weibull e a distribui\u00e7\u00e3o Gamma. S\u00e3o fornecidas as f\u00f3rmulas matem\u00e1ticas que definem cada uma dessas distribui\u00e7\u00f5es e s\u00e3o examinadas as implica\u00e7\u00f5es e as aplica\u00e7\u00f5es pr\u00e1ticas delas, tais como a avalia\u00e7\u00e3o da emiss\u00e3o de part\u00edculas em amostras radioativas ou o c\u00e1lculo da posi\u00e7\u00e3o de uma bola em um trilho com limites. Al\u00e9m disso, \u00e9 detalhado como essas distribui\u00e7\u00f5es podem ser modificadas e adaptadas mediante a aplica\u00e7\u00e3o de par\u00e2metros espec\u00edficos.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo final desta aula, o estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> o que s\u00e3o as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade.<\/li>\n<li><strong>Aplicar<\/strong> as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais conhecidas: exponencial, uniforme retangular, normal (gaussiana), Weibull e Gamma.<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>O que s\u00e3o as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>As 5 distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais conhecidas<\/strong><\/a><br \/>\n<a href=\"#3\">Distribui\u00e7\u00e3o Exponencial<\/a><br \/>\n<a href=\"#4\">Distribui\u00e7\u00e3o Uniforme Retangular<\/a><br \/>\n<a href=\"#5\">Distribui\u00e7\u00e3o Normal (Gaussiana)<\/a><br \/>\n<a href=\"#6\">Distribui\u00e7\u00e3o Weibull<\/a><br \/>\n<a href=\"#7\">Distribui\u00e7\u00e3o Gamma<\/a><br \/>\n<a href=\"#8\"><strong>Exerc\u00edcios<\/strong><\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/REOTUa7K8uQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Quando revisamos o relativo aos <a href=\"http:\/\/toposuranos.com\/material\/pt\/conheca-o-espaco-amostral-da-teoria-das-probabilidades\/\" target=\"_blank\" rel=\"noopener\">espa\u00e7os amostrais<\/a> vimos que estes podem ser de duas esp\u00e9cies: discretos e cont\u00ednuos. Tamb\u00e9m revisamos o que constitui uma <a href=\"http:\/\/toposuranos.com\/material\/pt\/distribuicoes-discretas-de-probabilidade-e-exemplos\/\" target=\"_blank\" rel=\"noopener\">distribui\u00e7\u00e3o de probabilidade discreta.<\/a> Agora \u00e9 a vez das distribui\u00e7\u00f5es cont\u00ednuas de probabilidade.<\/p>\n<p>&nbsp;<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>O que s\u00e3o as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade?<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=86s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Diremos que uma vari\u00e1vel aleat\u00f3ria<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem uma distribui\u00e7\u00e3o cont\u00ednua de probabilidade se existir uma fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X : \\mathbb{R} \\longrightarrow \\mathbb{R}^+,<\/span><\/span> que chamaremos <strong>Densidade de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,<\/span><\/span><\/strong> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall A \\subseteq \\mathbb{R}<\/span><\/span> valha a igualdade<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\in A) = \\displaystyle \\int_A f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Em particular, se tomarmos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=]a,b]<\/span><\/span> teremos<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(a\\lt X \\leq b) = \\displaystyle \\int_a^b f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">e se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=-\\infty<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P( X \\leq x) = \\displaystyle \\int_{-\\infty}^x f_X(t)dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">E al\u00e9m disso, a partir da propriedade (c) das <a href=\"http:\/\/toposuranos.com\/material\/pt\/variaveis-aleatorias-e-distribuicoes-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">distribui\u00e7\u00f5es de probabilidade<\/a> teremos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty} f_X(t)dt = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Aplicando o teorema fundamental do c\u00e1lculo sobre esta \u00faltima express\u00e3o, temos que para uma distribui\u00e7\u00e3o cont\u00ednua, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x),<\/span><\/span> \u00e9 cont\u00ednua para todos os <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,<\/span><\/span> e sua derivada \u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> para todos os valores <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> seja cont\u00ednua. Da continuidade de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x)<\/span><\/span> e da propriedade (d) (<a href=\"http:\/\/toposuranos.com\/material\/pt\/variaveis-aleatorias-e-distribuicoes-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">ver aqui<\/a>) deduz-se que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x=X)=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">E portanto<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x\\leq X)= P(x\\lt X)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 qualquer fun\u00e7\u00e3o que cumpre com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f\\geq 0<\/span><\/span> e com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}f(x)dx = 1,<\/span><\/span> ent\u00e3o se diz que \u00e9 uma densidade.<\/p>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>As 5 distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais conhecidas<\/h2>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>Distribui\u00e7\u00e3o Exponencial<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=714s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Uma fun\u00e7\u00e3o de distribui\u00e7\u00e3o exponencial<\/span><\/strong><\/a> com par\u00e2metro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\gt 0 <\/span><\/span> \u00e9 uma fun\u00e7\u00e3o de distribui\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> da forma.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{lll}\n\n1 - e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Em consequ\u00eancia, sua fun\u00e7\u00e3o de densidade \u00e9 da forma<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = \\left\\{\\begin{array}{lll}\n\n\\frac{1}{\\alpha}e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se uma vari\u00e1vel aleat\u00f3ria tem distribui\u00e7\u00e3o exponencial com par\u00e2metro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> escrevemos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">No contexto da distribui\u00e7\u00e3o de Poisson, se temos uma amostra radioativa que emite uma part\u00edcula com uma taxa m\u00e9dia de emiss\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span><\/span> ent\u00e3o o instante de tempo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> em que emite a primeira part\u00edcula tem distribui\u00e7\u00e3o exponencial com par\u00e2metro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1\/c.<\/span><\/span> Em outras palavras <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ex(1\/c),<\/span><\/span> e em consequ\u00eancia:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(T\\geq t)= e^{-ct}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>Distribui\u00e7\u00e3o Uniforme Retangular<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=930s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Uma distribui\u00e7\u00e3o uniforme retangular<\/span><\/strong><\/a> sobre um intervalo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u00e9 aquela que \u00e9 definida pela fun\u00e7\u00e3o de densidade<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\left\\{\\begin{array}{lll}\n\n\\displaystyle\\frac{1}{b-a} &amp; ; &amp; x\\in[a,b] \\\\ \\\\\n\n0 &amp; ; &amp; E.O.C.\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se soltarmos uma pequena bola em um trilho com limites nas extremidades do intervalo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b],<\/span><\/span> e esta rebater elasticamente ao chocar com as bordas, ent\u00e3o a vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> associada \u00e0 posi\u00e7\u00e3o de parada da bola por efeito do atrito tem distribui\u00e7\u00e3o uniforme retangular e se escreve <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Un(a,b)<\/span>.<\/span><\/p>\n<p><a name=\"5\"><\/a><\/br><\/br><\/p>\n<h3>Distribui\u00e7\u00e3o Normal (Gaussiana)<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1109s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Entre as distribui\u00e7\u00f5es cont\u00ednuas<\/span><\/strong><\/a> de probabilidade, a distribui\u00e7\u00e3o normal \u00e9 uma das mais populares na pr\u00e1tica.<\/p>\n<h4>Distribui\u00e7\u00e3o normal padr\u00e3o<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1150s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Define-se a densidade normal padr\u00e3o<\/span><\/strong><\/a> atrav\u00e9s da fun\u00e7\u00e3o<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\phi_{0,1}(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Por sua defini\u00e7\u00e3o, \u00e9 claro que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi\\gt 0.<\/span><\/span> Portanto, pode-se verificar que isso \u00e9 uma densidade de probabilidade simplesmente corroborando que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Esta \u00faltima igualdade pode ser demonstrada calculando o valor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2<\/span><\/span> quando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I =\\int_{-\\infty}^{+\\infty}\\phi(x)dx=1.<\/span><\/span> De fato, tem-se que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nI^2 &amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}dx \\\\ \\\\\n\n&amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-y^2\/2} dy \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{1}{{2\\pi}} \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy \\\\ \\\\\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Mas acontece que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy = \\int_{0}^{2\\pi} \\int_{0}^{+\\infty} e^{-r^2\/2} rdr d\\theta = 2\\pi <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Portanto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2 = 1,<\/span><\/span> de modo que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I=\\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx = 1. <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">A partir da densidade normal padr\u00e3o define-se a distribui\u00e7\u00e3o normal padr\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x) = \\int_{-\\infty}^x\\phi_{0,1}(t)dt.<\/span><\/span> Se uma vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o normal padr\u00e3o, ent\u00e3o se escreve <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(0,1).<\/span><\/span> A distribui\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x)<\/span><\/span> n\u00e3o pode ser calculada de forma expl\u00edcita, no entanto, existem tabelas que permitem obter rapidamente valores aproximados.<\/p>\n<h4>Distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span><\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A partir da densidade da distribui\u00e7\u00e3o normal padr\u00e3o<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_{0,1}<\/span><\/span> \u00e9 poss\u00edvel construir a densidade para a distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu\\in\\mathbb{R}<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma\\gt 0 <\/span><\/span> s\u00e3o, respectivamente, a m\u00e9dia e o desvio padr\u00e3o. A densidade da distribui\u00e7\u00e3o normal com esses par\u00e2metros fica escrita da seguinte maneira:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\phi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">De modo que a distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span>, fica da forma<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x\\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{t-\\mu}{\\sigma} \\right)dt = \\frac{1}{\\sqrt{2\\pi\\sigma}}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se a vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o normal com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu, \\sigma,<\/span><\/span> ent\u00e3o se escreve <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(\\mu, \\sigma).<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/br><\/br><\/p>\n<h3>Distribui\u00e7\u00e3o Weibull<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A distribui\u00e7\u00e3o Weibull<\/span><\/strong><\/a> com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\gt 0<\/span><\/span> tem uma fun\u00e7\u00e3o de distribui\u00e7\u00e3o da forma<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{llr}\n\n\\left(1 - e^{-t\/\\alpha} \\right)^\\beta &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se uma vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o Weibull com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> se escreve <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim We(\\alpha,\\beta).<\/span><\/span> A distribui\u00e7\u00e3o Weibull \u00e9 uma generaliza\u00e7\u00e3o para a distribui\u00e7\u00e3o exponencial, note que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">We(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/br><\/br><\/p>\n<h3>Distribui\u00e7\u00e3o Gamma<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2311s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A distribui\u00e7\u00e3o Gamma<\/span><\/strong><\/a> com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta,\\alpha<\/span><\/span> tem uma fun\u00e7\u00e3o de densidade da forma<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(t) = \\left\\{\\begin{array}{llr}\n\n\\displaystyle \\frac{1}{\\alpha \\Gamma(\\beta)}\\left(\\frac{t}{\\alpha} \\right)^{\\beta-1}e^{-t\/\\alpha} &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s) = \\displaystyle \\int_0^{+\\infty}u^{s-1}e^{-u}du <\/span><\/span> \u00e9 o que se conhece como \u00abFun\u00e7\u00e3o Gamma\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Uma das propriedades mais not\u00e1veis da fun\u00e7\u00e3o Gamma \u00e9 que permite generalizar os fatoriais dos n\u00fameros naturais sobre os reais (e at\u00e9 os complexos). N\u00e3o \u00e9 complicado verificar que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s+1) = s\\Gamma(s)<\/span><\/span> integrando por partes. Al\u00e9m disso, como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(1)=1<\/span><\/span> resulta que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall n\\in\\mathbb{N}\\right)\\left(\\Gamma(n) = (n-1)! \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se uma vari\u00e1vel aleat\u00f3ria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> tem distribui\u00e7\u00e3o Gamma com par\u00e2metros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta, \\alpha<\/span><\/span> se escreve <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ga(\\alpha,\\beta).<\/span><\/span> A distribui\u00e7\u00e3o Gamma \u00e9 outra generaliza\u00e7\u00e3o para a distribui\u00e7\u00e3o exponencial, note que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Ga(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Em um processo de Poisson com frequ\u00eancia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> (como um decaimento radioativo), se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> \u00e9 a vari\u00e1vel aleat\u00f3ria que representa o instante em que se produz o m-\u00e9simo evento; ent\u00e3o, dado um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\geq 0<\/span><\/span> e um n\u00famero <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span><\/span> de eventos que ocorrem no intervalo de tempo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[0,t]<\/span><\/span> teremos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\lt T \\leftrightarrow N\\lt m<\/span><\/span> e, como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N\\sim Po(ct),<\/span><\/span> tem-se:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1-F_T(t) = P(T\\gt t) = \\displaystyle \\sum_{k=0}^{m-1}Po(k; ct)=e^{-ct}\\sum_{k=0}^{m-1}\\frac{(ct)^k}{k!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">E portanto, se derivarmos isso descobriremos que a fun\u00e7\u00e3o de densidade \u00e9<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = ce^{-ct}\\frac{(ct)^{m-1}}{(m-1)!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">E portanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ga(1\/c, m).<\/span><\/span><\/p>\n<p><a name=\"8\"><\/a><\/br><\/br><\/p>\n<h2>Exerc\u00edcios<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Encontrar a constante <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\frac{c}{x^2+1}<\/span><\/span> \u00e9 uma densidade de probabilidade e calcular a correspondente fun\u00e7\u00e3o de distribui\u00e7\u00e3o de probabilidade (distribui\u00e7\u00e3o de Cauchy)<\/li>\n<li>A partir da fun\u00e7\u00e3o de densidade da distribui\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Un(a.b),<\/span><\/span> determinar sua correspondente fun\u00e7\u00e3o de distribui\u00e7\u00e3o.<\/li>\n<li>Demonstrar que a fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span> \u00e9 uma fun\u00e7\u00e3o de distribui\u00e7\u00e3o de probabilidade.<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kdxgrB1h98g\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Distribui\u00e7\u00f5es Cont\u00ednuas de Probabilidade ResumoAqui examinaremos em profundidade o conceito de distribui\u00e7\u00f5es cont\u00ednuas de probabilidade, destacando as caracter\u00edsticas e usos das cinco mais conhecidas: a distribui\u00e7\u00e3o exponencial, a distribui\u00e7\u00e3o uniforme retangular, a distribui\u00e7\u00e3o normal (Gaussiana), a distribui\u00e7\u00e3o Weibull e a distribui\u00e7\u00e3o Gamma. S\u00e3o fornecidas as f\u00f3rmulas matem\u00e1ticas que definem cada uma dessas distribui\u00e7\u00f5es e s\u00e3o [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26864,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":25,"footnotes":""},"categories":[571,672],"tags":[],"class_list":["post-26870","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica-pt","category-probabilidades-e-estatistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais importantes: exponencial, uniforme, normal, Weibull e Gamma.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/pt\/distribuicoes-continuas-de-probabilidade\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade\" \/>\n<meta property=\"og:description\" content=\"Descubra as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais importantes: exponencial, uniforme, normal, Weibull e Gamma.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/distribuicoes-continuas-de-probabilidade\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-05-25T22:48:04+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-25T22:49:15+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/distribucionescontinuas.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade\" \/>\n<meta name=\"twitter:description\" content=\"Descubra as distribui\u00e7\u00f5es cont\u00ednuas de probabilidade mais importantes: exponencial, uniforme, normal, Weibull e Gamma.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/distribucionescontinuas.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade\",\"datePublished\":\"2024-05-25T22:48:04+00:00\",\"dateModified\":\"2024-05-25T22:49:15+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/\"},\"wordCount\":1834,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/distribucionescontinuas.jpg\",\"articleSection\":[\"Matem\u00e1tica\",\"Probabilidades e Estat\u00edstica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/distribuicoes-continuas-de-probabilidade\\\/\",\"name\":\"Distribui\u00e7\u00f5es cont\u00ednuas de probabilidade - 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