{"id":231,"date":"2023-08-03T20:59:59","date_gmt":"2023-08-03T20:59:59","guid":{"rendered":"https:\/\/statorials.org\/pt\/distribuicao-binomial-negativa-1\/"},"modified":"2023-08-03T20:59:59","modified_gmt":"2023-08-03T20:59:59","slug":"distribuicao-binomial-negativa-1","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/distribuicao-binomial-negativa-1\/","title":{"rendered":"Distribui\u00e7\u00e3o binomial negativa"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o binomial negativa e para que ela \u00e9 usada. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula da distribui\u00e7\u00e3o binomial negativa, um exemplo concreto e as propriedades deste tipo de distribui\u00e7\u00e3o de probabilidade. Finalmente, voc\u00ea poder\u00e1 calcular qualquer probabilidade de distribui\u00e7\u00e3o binomial negativa com uma calculadora online. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-binomial-negativa\"><\/span> Qual \u00e9 a distribui\u00e7\u00e3o binomial negativa?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>distribui\u00e7\u00e3o binomial negativa<\/strong> \u00e9 uma distribui\u00e7\u00e3o de probabilidade que descreve o n\u00famero de tentativas de Bernoulli necess\u00e1rias para obter um determinado n\u00famero de resultados positivos.<\/p>\n<p> Portanto, uma distribui\u00e7\u00e3o binomial negativa tem dois par\u00e2metros caracter\u00edsticos: <em>r<\/em> \u00e9 o n\u00famero de resultados desejados e <em>p<\/em> \u00e9 a probabilidade de sucesso para cada experimento de Bernoulli realizado.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-171122de529a1c006bc46e8d89176016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim \\text{BN}(r,p)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Lembre-se de que um teste de Bernoulli \u00e9 um experimento que tem dois resultados poss\u00edveis: \u201csucesso\u201d e \u201cfracasso\u201d. Portanto, se a probabilidade de \u201csucesso\u201d for <em>p<\/em> , a probabilidade de \u201cfracasso\u201d \u00e9 <em>q=1-p<\/em> .<\/p>\n<p> Assim, uma distribui\u00e7\u00e3o binomial negativa define um processo no qual s\u00e3o realizadas quantas tentativas de Bernoulli forem necess\u00e1rias para obter <em>resultados<\/em> positivos. Al\u00e9m disso, todos estes ensaios de Bernoulli s\u00e3o independentes e t\u00eam uma probabilidade constante de <em>sucesso<\/em> .<\/p>\n<p> Por exemplo, uma vari\u00e1vel aleat\u00f3ria que segue uma distribui\u00e7\u00e3o binomial negativa \u00e9 o n\u00famero de vezes que um dado deve ser lan\u00e7ado at\u00e9 que o n\u00famero 6 seja lan\u00e7ado tr\u00eas vezes.<\/p>\n<p> A diferen\u00e7a entre uma distribui\u00e7\u00e3o binomial negativa e uma distribui\u00e7\u00e3o binomial \u00e9 que a distribui\u00e7\u00e3o binomial negativa conta o n\u00famero de vezes que leva para obter um certo n\u00famero de resultados bem-sucedidos, enquanto a distribui\u00e7\u00e3o binomial conta o n\u00famero de casos bem-sucedidos em uma s\u00e9rie de testes de Bernoulli. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/statorials.org\/pt\/distribuicao-binomial-1\/\">Qual \u00e9 a distribui\u00e7\u00e3o binomial?<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-distribucion-binomial-negativa\"><\/span> F\u00f3rmula de distribui\u00e7\u00e3o binomial negativa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dados os par\u00e2metros <em>r, p, x,<\/em> a probabilidade de uma distribui\u00e7\u00e3o binomial negativa \u00e9 calculada multiplicando o n\u00famero combinat\u00f3rio de <em>x-1<\/em> em <em>xr<\/em> por <em>(1-p) <sup>xr<\/sup><\/em> por <em>p <sup>r<\/sup><\/em> .<\/p>\n<p> Portanto, a <strong>f\u00f3rmula para calcular uma probabilidade de distribui\u00e7\u00e3o binomial negativa<\/strong> \u00e9: <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/formule-de-distribution-binomiale-negative.png\" alt=\"F\u00f3rmula de distribui\u00e7\u00e3o binomial negativa\" class=\"wp-image-4813\" width=\"318\" height=\"319\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> \ud83d\udc49 <u style=\"text-decoration-color:#FF8A05;\">Voc\u00ea pode usar a calculadora abaixo para calcular a probabilidade de uma vari\u00e1vel que segue a distribui\u00e7\u00e3o binomial negativa.<\/u> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicio-resuelto-de-la-distribucion-binomial-negativa\"><\/span> Exerc\u00edcio resolvido da distribui\u00e7\u00e3o binomial negativa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> Qual \u00e9 a probabilidade de que, se voc\u00ea lan\u00e7ar uma moeda oito vezes, ela d\u00ea cara pela quarta vez no oitavo lan\u00e7amento?<\/li>\n<\/ul>\n<p> Primeiro, precisamos calcular a probabilidade de obter cara no lan\u00e7amento de uma moeda. Neste caso, temos apenas um resultado positivo (cara) entre dois resultados poss\u00edveis (cara e coroa), ent\u00e3o a probabilidade de sucesso \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bdceac6409b69d142f23801ec85e2691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p=\\cfrac{1}{2}=0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"93\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Assim, a vari\u00e1vel aleat\u00f3ria neste problema segue uma distribui\u00e7\u00e3o binomial negativa onde r=4 ep=0,5. Portanto, usamos a f\u00f3rmula da distribui\u00e7\u00e3o binomial negativa para calcular a probabilidade que o exerc\u00edcio nos pede para fazer. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bc56100604e5889a6d169c0395f19ebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P[X=x]&amp;=\\begin{pmatrix}x-1\\\\ x-r\\end{pmatrix}\\cdot (1-p)^{x-r}\\cdot p^r\\\\[2ex]\\displaystyle P[X=8]&amp;=\\begin{pmatrix}8-1\\\\ 8-4\\end{pmatrix}\\cdot (1-0,5)^{8-4}\\cdot 0,5^4\\\\[2ex] P[X=8]&amp;=0,1367\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"316\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-la-distribucion-binomial-negativa\"><\/span> Caracter\u00edsticas da distribui\u00e7\u00e3o binomial negativa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Abaixo est\u00e3o as caracter\u00edsticas mais importantes da distribui\u00e7\u00e3o binomial negativa.<\/p>\n<ul>\n<li> A distribui\u00e7\u00e3o binomial negativa \u00e9 definida por dois par\u00e2metros caracter\u00edsticos: <em>r<\/em> \u00e9 o n\u00famero de resultados desejados com sucesso e <em>p<\/em> \u00e9 a probabilidade de sucesso para cada experimento de Bernoulli realizado.<\/li>\n<\/ul>\n<pre class=\"ql-errors\"> *** QuickLaTeX cannot compile formula:\n\\begin{array}{c}r\\in \\mathbb{Z}^+ \\\\[2ex] 0 &lt;ul&gt;&lt;li&gt; The mean of the negative binomial distribution is equal to &lt;em&gt;r&lt;\/em&gt; multiplied by &lt;em&gt;(1-p)&lt;\/em&gt; and divided by &lt;em&gt;p&lt;\/em&gt; . Thus the formula which makes it possible to calculate the mean of a negative binomial distribution is the following: &lt;\/li&gt;&lt;\/ul&gt;[latex]E[X]=\\cfrac{r\\cdot (1-p)}{p}\n\n*** Error message:\nMissing $ inserted.\nleading text: \\begin{array}{c}\nPlease use \\mathaccent for accents in math mode.\nleading text: ...The mean of the binomial distribution born\nPlease use \\mathaccent for accents in math mode.\nleading text: ... the negative binomial distribution is\nPlease use \\mathaccent for accents in math mode.\nleading text: ...negative binomial distribution is equal to\nPlease use \\mathaccent for accents in math mode.\nleading text: ...gative is equal to &lt;em&gt;r&lt;\/em&gt; multiplied\nPlease use \\mathaccent for accents in math mode.\nleading text: ...m&gt; multiplied by &lt;em&gt;(1-p)&lt;\/em&gt; and divided\nPlease use \\mathaccent for accents in math mode.\nleading text: ...the mean of a binomial distribution born\n\\begin{array} on input line 8 ended by \\end{document}.\n\n<\/pre>\n<ul>\n<li> A vari\u00e2ncia de uma distribui\u00e7\u00e3o binomial negativa \u00e9 igual a <em>r<\/em> multiplicado por <em>(1-p)<\/em> dividido por <em>p <sup>2<\/sup><\/em> .<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-c82d66018d0715f20318f383cf7c04c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Var(X)=\\cfrac{r\\cdot (1-p)}{p^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"163\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> Se o par\u00e2metro <em>r<\/em> for maior que 1, a moda de uma distribui\u00e7\u00e3o binomial negativa pode ser calculada com a seguinte f\u00f3rmula:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-600d3f94b3cd3927f264450bcd043a86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lfloor \\frac{(r-1)(1-p)}{p}\\rfloor\\quad \\text{para }r>1&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;42&#8243; width=&#8221;221&#8243; style=&#8221;vertical-align: -16px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> A fun\u00e7\u00e3o de massa que permite determinar a probabilidade de uma distribui\u00e7\u00e3o binomial negativa \u00e9 a seguinte:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-beafc8c08e7aad51059349dfd2addd5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P[X=x]=\\begin{pmatrix}x-1\\\\ x-r\\end{pmatrix}\\cdot (1-p)^{x-r}\\cdot p^r\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"284\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> O coeficiente de assimetria de uma distribui\u00e7\u00e3o binomial negativa \u00e9 calculado com a seguinte express\u00e3o:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-9bc0a97cdf77d4dd439c71cf8637e53c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\frac{2-p}{\\sqrt{r\\,(1-p)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"122\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<ul>\n<li> A curtose de uma distribui\u00e7\u00e3o binomial negativa pode ser encontrada com a seguinte f\u00f3rmula:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-510199601f8ac60aa64008f00d959562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C=\\frac{6}{r} + \\frac{p^2}{r\\,(1-p)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"139\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> Se o par\u00e2metro <em>r<\/em> for igual a 1, ent\u00e3o temos um caso de <a href=\"https:\/\/statorials.org\/pt\/distribuicao-geometrica\/\">distribui\u00e7\u00e3o geom\u00e9trica<\/a> . <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-24bb5ef017a69a1a9de1e07f32412fcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r=1 \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad X\\sim \\text{Geometrica}(p)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"350\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calculadora-de-la-distribucion-binomial-negativa\"><\/span> Calculadora de distribui\u00e7\u00e3o binomial negativa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Insira os valores dos par\u00e2metros <em>r, p, x<\/em> na calculadora a seguir para calcular a probabilidade. Voc\u00ea deve inserir n\u00fameros usando o ponto como separador decimal, por exemplo 0,50.<\/p>\n<form action=\"\" method=\"post\">\n<div style=\"margin-bottom:20px; margin-left:5%\"> <span style=\"color:#101010;font-weight: normal;\"><span style=\"color:#1C2C92\"><strong>\u27a4<\/strong><\/span> N\u00famero de resultados bem-sucedidos <span style=\"color:#1C2C92\"><strong>\u2192<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-2060a909f2905bcbe83f8fe9152c2284_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r = \" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> <input name=\"r\" style=\"border:1.5px solid #4FC3F7; border-radius:5px;  padding:7px; color:#000000; background-color:#EBF5FB; width:60px\" required=\"\" oninvalid=\"this.setCustomValidity('Introduce el valor de r aqu\u00ed')\" oninput=\"this.setCustomValidity('')\"><\/div>\n<div style=\"margin-bottom:20px; margin-left:5%\"> <span style=\"color:#101010;font-weight: normal;\"><span style=\"color:#1C2C92\"><strong>\u27a4<\/strong><\/span> Probabilidade de sucesso de cada experimento <span style=\"color:#1C2C92\"><strong>\u2192<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e67f693705a2e492d8981f0ed1387c3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"p = \" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> <input name=\"prob\" style=\"border:1.5px solid #4FC3F7; border-radius:5px;  padding:7px; color:#000000; background-color:#EBF5FB; width:60px\" required=\"\" oninvalid=\"this.setCustomValidity('Introduce el valor de p aqu\u00ed')\" oninput=\"this.setCustomValidity('')\"><\/div>\n<div style=\"margin-bottom:20px; margin-left:5%\"> <span style=\"color:#101010;font-weight: normal;\"><span style=\"color:#1C2C92\"><strong>\u27a4<\/strong><\/span> N\u00famero total de experimentos realizados <span style=\"color:#1C2C92\"><strong>\u2192<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-04ae6e91e4423afa96cc9b717c007c88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> <input name=\"xigual\" style=\"border:1.5px solid #4FC3F7; border-radius:5px;  padding:7px; color:#000000; background-color:#EBF5FB; width:60px\" required=\"\" oninvalid=\"this.setCustomValidity('Introduce el valor de x aqu\u00ed')\" oninput=\"this.setCustomValidity('')\"><\/div>\n<div style=\"text-align:center\"><input align=\"center\" style=\"border-radius:30px; margin: 20px\" type=\"submit\" name=\"submit\" value=\"Calcule a probabilidade\"><\/div>\n<\/form>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o binomial negativa e para que ela \u00e9 usada. Voc\u00ea tamb\u00e9m encontrar\u00e1 a f\u00f3rmula da distribui\u00e7\u00e3o binomial negativa, um exemplo concreto e as propriedades deste tipo de distribui\u00e7\u00e3o de probabilidade. Finalmente, voc\u00ea poder\u00e1 calcular qualquer probabilidade de distribui\u00e7\u00e3o binomial negativa com uma calculadora online. Qual \u00e9 a distribui\u00e7\u00e3o [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-231","post","type-post","status-publish","format-standard","hentry","category-probabilidade"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Distribui\u00e7\u00e3o Binomial Negativa - Probabilidade e Estat\u00edstica<\/title>\n<meta name=\"description\" content=\"Este artigo explica o que \u00e9 a distribui\u00e7\u00e3o binomial negativa e para que ela \u00e9 usada. 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