{"id":222,"date":"2023-08-03T23:58:33","date_gmt":"2023-08-03T23:58:33","guid":{"rendered":"https:\/\/statorials.org\/pl\/rozklad-gamma\/"},"modified":"2023-08-03T23:58:33","modified_gmt":"2023-08-03T23:58:33","slug":"rozklad-gamma","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/rozklad-gamma\/","title":{"rendered":"Rozk\u0142ad gamma"},"content":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest rozk\u0142ad gamma i do czego si\u0119 go u\u017cywa. W ten spos\u00f3b znajdziesz definicj\u0119 rozk\u0142adu gamma, jego w\u0142a\u015bciwo\u015bci i jak wygl\u0105da jego graficzna reprezentacja. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-gamma\"><\/span> Jaki jest rozk\u0142ad gamma?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Rozk\u0142ad gamma<\/strong> jest ci\u0105g\u0142ym rozk\u0142adem prawdopodobie\u0144stwa zdefiniowanym przez dwa charakterystyczne parametry, \u03b1 i \u03bb. Inaczej m\u00f3wi\u0105c, rozk\u0142ad gamma zale\u017cy od warto\u015bci jego dw\u00f3ch parametr\u00f3w: \u03b1 jest parametrem kszta\u0142tu, a \u03bb jest parametrem skali.<\/p>\n<p> Symbolem rozk\u0142adu gamma jest wielka grecka litera \u0393. Zatem, je\u015bli zmienna losowa ma rozk\u0142ad gamma, zapisuje si\u0119 j\u0105 w nast\u0119puj\u0105cy spos\u00f3b:<\/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-1b0ab2e724ffd74455d0907b39f4a598_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim \\Gamma(\\alpha,\\lambda)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Rozk\u0142ad gamma mo\u017cna r\u00f3wnie\u017c parametryzowa\u0107 za pomoc\u0105 parametru kszta\u0142tu k = \u03b1 i parametru odwrotnej skali \u03b8 = 1\/\u03bb. We wszystkich przypadkach dwa parametry definiuj\u0105ce rozk\u0142ad gamma s\u0105 dodatnimi liczbami rzeczywistymi.<\/p>\n<p> Zazwyczaj rozk\u0142ad gamma jest u\u017cywany do modelowania zbior\u00f3w danych sko\u015bnych w prawo, dzi\u0119ki czemu po lewej stronie wykresu wyst\u0119puje wi\u0119ksza koncentracja danych. Na przyk\u0142ad rozk\u0142ad gamma s\u0142u\u017cy do modelowania niezawodno\u015bci komponent\u00f3w elektrycznych.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"grafica-de-la-distribucion-gamma\"><\/span> Schemat rozk\u0142adu gamma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Wykres rozk\u0142adu gamma zale\u017cy od warto\u015bci jego charakterystycznych parametr\u00f3w. Poni\u017cej mo\u017cesz zobaczy\u0107, jak zmienia si\u0119 funkcja g\u0119sto\u015bci rozk\u0142adu gamma w zale\u017cno\u015bci od parametru kszta\u0142tu i parametru skali. <\/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\/trace-de-distribution-gamma.png\" alt=\"wykres rozk\u0142adu gamma\" class=\"wp-image-4536\" width=\"648\" height=\"388\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Z drugiej strony, poni\u017cej mo\u017cesz zobaczy\u0107 wykres skumulowanej funkcji prawdopodobie\u0144stwa rozk\u0142adu gamma: <\/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-gamma-probabilite-cumulative.png\" alt=\"wykres skumulowanej funkcji prawdopodobie\u0144stwa rozk\u0142adu Gamma\" class=\"wp-image-4537\" width=\"648\" height=\"388\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-la-distribucion-gamma\"><\/span> Charakterystyka rozk\u0142adu gamma<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Zobaczymy wtedy, jakie s\u0105 cechy rozk\u0142adu gamma.<\/p>\n<ul>\n<li> Wykres rozk\u0142adu gamma jest ca\u0142kowicie zdefiniowany przez jego dwa charakterystyczne parametry: \u03b1 jest parametrem kszta\u0142tu, a \u03bb jest parametrem skali.<\/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-9640d41832cbcef7d0a7791c55ace57e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha , \\lambda >0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;16&#8243; width=&#8221;62&#8243; style=&#8221;vertical-align: -4px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> Dziedzina rozk\u0142adu gamma sk\u0142ada si\u0119 wy\u0142\u0105cznie z liczb dodatnich.<\/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-f543506f97e1f9c5a56ccc4566a3febf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\in (0,+\\infty)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> \u015arednia rozk\u0142adu gamma jest r\u00f3wna stosunkowi parametru kszta\u0142tu do parametru skali, tj. \u03b1\/\u03bb.<\/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-326373413d0fe34ef96959dabe9c5cbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"E[X]=\\cfrac{\\alpha}{\\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"78\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<ul>\n<li> Wariancja rozk\u0142adu gamma jest r\u00f3wna parametrowi kszta\u0142tu podzielonemu przez kwadrat parametru skali.<\/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-693a549de92268ac21add32944199974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Var(X)=\\cfrac{\\alpha}{\\lambda^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<ul>\n<li> Dla warto\u015bci \u03b1 mniejszych ni\u017c 1 tryb wynosi 0. Je\u015bli jednak \u03b1 jest r\u00f3wne lub wi\u0119ksze ni\u017c 1, tryb rozk\u0142adu gamma mo\u017cna obliczy\u0107 za pomoc\u0105 nast\u0119puj\u0105cego wzoru:<\/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-c9e2762af62f9c38870902ec0284ee07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}Mo=0 \\qquad \\text{para } \\alpha<1\\\\[2ex]Mo=\\cfrac{\\alpha-1}{\\lambda} \\qquad \\text{para } \\alpha\\geq1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"214\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Wz\u00f3r na funkcj\u0119 g\u0119sto\u015bci rozk\u0142adu gamma to:<\/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-41689ffee11808eab10064a7f3a77693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{\\lambda(\\lambda x)^{\\alpha-1}e^{-\\lambda x}}{\\Gamma(\\alpha)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"168\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Gdzie \u0393 jest funkcj\u0105 gamma zdefiniowan\u0105 jako:<\/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-6e244faebd1174cb6a56e4254bdaaec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\Gamma(\\alpha)=\\int_0^\\infty t^{\\alpha-1}e^{-t}dt\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"169\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> Wz\u00f3r na skumulowany rozk\u0142ad zmiennej losowej okre\u015blonej rozk\u0142adem gamma jest nast\u0119puj\u0105cy:<\/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-6bc3247a247daec5048c409319dc3943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle F(x)=\\int_0^x\\frac{\\lambda(\\lambda y)^{\\alpha-1}e^{-\\lambda y}}{\\Gamma(\\alpha)}\\;dy\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"224\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> Je\u015bli parametr kszta\u0142tu \u03b1 jest r\u00f3wny 1, w\u00f3wczas rozk\u0142ad gamma jest r\u00f3wnowa\u017cny rozk\u0142adowi wyk\u0142adniczemu z tym samym parametrem skali \u03bb.<\/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-60ed376f92f0db2c22e19f85bb76f828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim \\Gamma(1,\\lambda) \\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ X\\sim \\text{Exp}(\\lambda)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"321\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Gdy parametr skali \u03bb jest \u015bredni\u0105, w\u00f3wczas rozk\u0142ad gamma jest szczeg\u00f3lnym przypadkiem <a href=\"https:\/\/statorials.org\/pl\/rozk\u0142ad-chi-kwadrat\/\">rozk\u0142adu chi-kwadrat<\/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-efb733464c432d7d1b22ed7e5c534979_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X\\sim \\Gamma\\left(\\frac{n}{2},\\frac{1}{2}\\right) \\text{con } n\\in \\mathbb{N}\\ \\color{orange}\\bm{\\longrightarrow}\\color{black} \\ X\\sim \\chi_n^2\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"389\" style=\"vertical-align: -17px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>W tym artykule wyja\u015bniono, czym jest rozk\u0142ad gamma i do czego si\u0119 go u\u017cywa. W ten spos\u00f3b znajdziesz definicj\u0119 rozk\u0142adu gamma, jego w\u0142a\u015bciwo\u015bci i jak wygl\u0105da jego graficzna reprezentacja. Jaki jest rozk\u0142ad gamma? Rozk\u0142ad gamma jest ci\u0105g\u0142ym rozk\u0142adem prawdopodobie\u0144stwa zdefiniowanym przez dwa charakterystyczne parametry, \u03b1 i \u03bb. Inaczej m\u00f3wi\u0105c, rozk\u0142ad gamma zale\u017cy od warto\u015bci jego [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-222","post","type-post","status-publish","format-standard","hentry","category-prawdopodobienstwo"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Rozk\u0142ad gamma<\/title>\n<meta name=\"description\" content=\"Tutaj dowiesz si\u0119 czym jest rozk\u0142ad gamma, do czego si\u0119 go u\u017cywa, jego wykres i charakterystyk\u0119 rozk\u0142adu gamma.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/statorials.org\/pl\/rozklad-gamma\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Rozk\u0142ad gamma\" \/>\n<meta property=\"og:description\" content=\"Tutaj dowiesz si\u0119 czym jest rozk\u0142ad gamma, do czego si\u0119 go u\u017cywa, jego wykres i charakterystyk\u0119 rozk\u0142adu gamma.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/rozklad-gamma\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-03T23:58:33+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-1b0ab2e724ffd74455d0907b39f4a598_l3.png\" \/>\n<meta name=\"author\" content=\"Benjamin Anderson\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Napisane przez\" \/>\n\t<meta name=\"twitter:data1\" content=\"Benjamin Anderson\" \/>\n\t<meta name=\"twitter:label2\" content=\"Szacowany czas czytania\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minuty\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/rozklad-gamma\/\",\"url\":\"https:\/\/statorials.org\/pl\/rozklad-gamma\/\",\"name\":\"\u25b7 Rozk\u0142ad gamma\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-08-03T23:58:33+00:00\",\"dateModified\":\"2023-08-03T23:58:33+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Tutaj dowiesz si\u0119 czym jest rozk\u0142ad gamma, do czego si\u0119 go u\u017cywa, jego wykres i charakterystyk\u0119 rozk\u0142adu gamma.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/rozklad-gamma\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/rozklad-gamma\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/rozklad-gamma\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Rozk\u0142ad gamma\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/statorials.org\/pl\/#website\",\"url\":\"https:\/\/statorials.org\/pl\/\",\"name\":\"Statorials\",\"description\":\"Tw\u00f3j przewodnik po kompetencjach statystycznych!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/statorials.org\/pl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pl-PL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\",\"name\":\"Benjamin Anderson\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pl-PL\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"contentUrl\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"caption\":\"Benjamin Anderson\"},\"description\":\"Cze\u015b\u0107, jestem Benjamin i jestem emerytowanym profesorem statystyki, kt\u00f3ry zosta\u0142 oddanym nauczycielem Statorials. 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