{"id":214,"date":"2023-08-04T02:39:23","date_gmt":"2023-08-04T02:39:23","guid":{"rendered":"https:\/\/statorials.org\/pt\/distribuicao-beta\/"},"modified":"2023-08-04T02:39:23","modified_gmt":"2023-08-04T02:39:23","slug":"distribuicao-beta","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/distribuicao-beta\/","title":{"rendered":"Distribui\u00e7\u00e3o beta"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 distribui\u00e7\u00e3o beta e para que \u00e9 usada. Da mesma forma, voc\u00ea poder\u00e1 ver o gr\u00e1fico da distribui\u00e7\u00e3o beta e as propriedades deste tipo de distribui\u00e7\u00e3o de probabilidade. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-beta\"><\/span> Qual \u00e9 a distribui\u00e7\u00e3o beta?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>distribui\u00e7\u00e3o beta<\/strong> \u00e9 uma distribui\u00e7\u00e3o de probabilidade definida no intervalo (0,1) e parametrizada por dois par\u00e2metros positivos: \u03b1 e \u03b2. Em outras palavras, os valores da distribui\u00e7\u00e3o beta dependem dos par\u00e2metros \u03b1 e \u03b2.<\/p>\n<p> Portanto, a principal caracter\u00edstica da distribui\u00e7\u00e3o beta \u00e9 que sua forma pode ser controlada pelos par\u00e2metros \u03b1 e \u03b2. Al\u00e9m disso, a distribui\u00e7\u00e3o beta \u00e9 usada para definir vari\u00e1veis aleat\u00f3rias cujo valor est\u00e1 entre 0 e 1.<\/p>\n<p> Existem diversas nota\u00e7\u00f5es para indicar que uma vari\u00e1vel aleat\u00f3ria cont\u00ednua \u00e9 governada por uma distribui\u00e7\u00e3o beta, as mais comuns s\u00e3o:<\/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-ee1d0d8a1624a017b8ef9ce8a67c694e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}X\\sim B(\\alpha,\\beta)\\\\[2ex]X\\sim Beta(\\alpha,\\beta)\\\\[2ex]X\\sim \\beta_{\\alpha,\\beta}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"121\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Nas estat\u00edsticas, a distribui\u00e7\u00e3o beta tem aplica\u00e7\u00f5es muito variadas. Por exemplo, a distribui\u00e7\u00e3o beta \u00e9 usada para estudar varia\u00e7\u00f5es de porcentagens em diferentes amostras. Da mesma forma, no gerenciamento de projetos, a distribui\u00e7\u00e3o beta \u00e9 usada para realizar an\u00e1lises Pert. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"grafica-de-la-distribucion-beta\"><\/span> Gr\u00e1fico de distribui\u00e7\u00e3o beta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Considerando a defini\u00e7\u00e3o de distribui\u00e7\u00e3o beta, a fun\u00e7\u00e3o de densidade e a fun\u00e7\u00e3o de distribui\u00e7\u00e3o de probabilidade da distribui\u00e7\u00e3o beta s\u00e3o plotadas abaixo.<\/p>\n<p> Abaixo voc\u00ea pode ver como o gr\u00e1fico da fun\u00e7\u00e3o densidade da distribui\u00e7\u00e3o beta varia dependendo dos par\u00e2metros \u03b1 e \u03b2. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-graphique-beta.png\" alt=\"gr\u00e1fico de distribui\u00e7\u00e3o beta\" class=\"wp-image-4315\" width=\"621\" height=\"372\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Da mesma forma, abaixo voc\u00ea pode ver a representa\u00e7\u00e3o gr\u00e1fica da probabilidade cumulativa da distribui\u00e7\u00e3o beta com base nos par\u00e2metros \u03b1 e \u03b2. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-beta-cumulative.png\" alt=\"gr\u00e1fico de distribui\u00e7\u00e3o beta cumulativa\" class=\"wp-image-4316\" width=\"621\" height=\"373\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-la-distribucion-beta\"><\/span> Caracter\u00edsticas da distribui\u00e7\u00e3o beta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nesta se\u00e7\u00e3o veremos quais s\u00e3o as caracter\u00edsticas mais importantes da distribui\u00e7\u00e3o beta.<\/p>\n<ul>\n<li> Os par\u00e2metros \u03b1 e \u03b2 da distribui\u00e7\u00e3o beta s\u00e3o n\u00fameros reais e positivos.<\/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-442bebaba847543d6db71cdf0127a4b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\alpha >0\\\\[2ex] \\beta >0\\end{array}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;54&#8243; width=&#8221;44&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> O dom\u00ednio da distribui\u00e7\u00e3o beta varia de 0 a 1, os dois extremos n\u00e3o est\u00e3o inclu\u00eddos.<\/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-60f8667881e2c5a08eae2db3c3e04708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\in (0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A m\u00e9dia da distribui\u00e7\u00e3o beta \u00e9 igual a alfa dividido pela soma alfa mais beta.<\/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-4530a8b72211111e8bec79b6388e00a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}X\\sim B(\\alpha,\\beta)\\\\[2ex] E[X]=\\cfrac{\\alpha}{\\alpha+\\beta}\\end{array} \" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"111\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> A vari\u00e2ncia da distribui\u00e7\u00e3o beta pode ser calculada usando 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-71c6ef40db833722e63c56cd763d6601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}X\\sim B(\\alpha,\\beta)\\\\[2ex] Var(X)=\\cfrac{\\alpha\\cdot \\beta}{(\\alpha+\\beta+1)\\cdot (\\alpha+\\beta)^2}\\end{array} \" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"255\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Para valores de alfa e beta maiores que 1, o modo de distribui\u00e7\u00e3o beta pode ser facilmente encontrado 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-472cafc0cf9d1cf782206ddc7c9475d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Mo=\\cfrac{\\alpha-1}{\\alpha+\\beta-2}\\qquad \\alpha,\\beta>1&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;42&#8243; width=&#8221;225&#8243; style=&#8221;vertical-align: -16px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> A fun\u00e7\u00e3o densidade da distribui\u00e7\u00e3o beta \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-48f1f2a13df23d7c779bc37da4b49368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P[X=x]=\\frac{x^{\\alpha-1}(1-x)^{\\beta-1}}{B(\\alpha,\\beta)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"218\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Onde B(\u03b1,\u03b2) \u00e9 a fun\u00e7\u00e3o beta, que \u00e9 definida como:<\/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-f4eb1d2659d9993f9d05a35cce63fc14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B(\\alpha,\\beta)=\\int_0^1x^{\\alpha-1}(1-x)^{\\beta-1}dx\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"248\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> A fun\u00e7\u00e3o de probabilidade cumulativa da distribui\u00e7\u00e3o beta \u00e9:<\/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-031b21ea3e2964d3ac4e6f222356d55b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P[X\\leq x]=\\frac{B(x;\\alpha,\\beta)}{B(\\alpha,\\beta)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"176\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Onde B(x;\u03b1,\u03b2) \u00e9 a fun\u00e7\u00e3o beta incompleta, definida como:<\/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-6c64764f48f86e8f97df5ed1e8d7fbe6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B(x;\\,a,b) = \\int_0^x t^{a-1}\\,(1-t)^{b-1}\\,dt\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"255\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> Se X \u00e9 uma vari\u00e1vel definida por uma distribui\u00e7\u00e3o beta, ent\u00e3o 1-X \u00e9 uma vari\u00e1vel definida por uma distribui\u00e7\u00e3o beta cujos par\u00e2metros alfa e beta s\u00e3o os par\u00e2metros beta e alfa da distribui\u00e7\u00e3o beta original, respectivamente.<\/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-c65dd399bb92ffe982784a6c1b6e5b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim B(\\alpha,\\beta) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ 1-X\\sim B(\\beta,\\alpha)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"361\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Se os par\u00e2metros alfa e beta da distribui\u00e7\u00e3o beta forem iguais a 1, ent\u00e3o a distribui\u00e7\u00e3o ser\u00e1 equivalente a uma distribui\u00e7\u00e3o uniforme dos par\u00e2metros 0 e 1.<\/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-10010ad6c6959b44a8ae0fe4361b4034_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim B(1,1) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ X\\sim U(0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"321\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 distribui\u00e7\u00e3o beta e para que \u00e9 usada. Da mesma forma, voc\u00ea poder\u00e1 ver o gr\u00e1fico da distribui\u00e7\u00e3o beta e as propriedades deste tipo de distribui\u00e7\u00e3o de probabilidade. Qual \u00e9 a distribui\u00e7\u00e3o beta? A distribui\u00e7\u00e3o beta \u00e9 uma distribui\u00e7\u00e3o de probabilidade definida no intervalo (0,1) e parametrizada por dois par\u00e2metros [&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-214","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 Beta - Probabilidades e Estat\u00edsticas<\/title>\n<meta name=\"description\" content=\"Este artigo explica o que \u00e9 distribui\u00e7\u00e3o beta e para que \u00e9 usada. 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