{"id":3935,"date":"2023-07-14T16:01:08","date_gmt":"2023-07-14T16:01:08","guid":{"rendered":"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/"},"modified":"2023-07-14T16:01:08","modified_gmt":"2023-07-14T16:01:08","slug":"wartosc-oczekiwana-x3","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/","title":{"rendered":"Jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 x^3"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\">W przypadku <a href=\"https:\/\/statorials.org\/pl\/zmienne-losowe\/\" target=\"_blank\" rel=\"noopener\">zmiennej losowej<\/a> oznaczonej X mo\u017cna zastosowa\u0107 nast\u0119puj\u0105cy wz\u00f3r do obliczenia oczekiwanej warto\u015bci X <sup>3<\/sup> :<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>E(X <sup>3<\/sup> ) = \u03a3x <sup>3<\/sup> * p(x)<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Z\u0142oto:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>\u03a3<\/strong> : Symbol oznaczaj\u0105cy \u201esum\u0119\u201d<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>x<\/strong> : Warto\u015b\u0107 zmiennej losowej<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p(x)<\/strong> : Prawdopodobie\u0144stwo, \u017ce zmienna losowa przyjmie dan\u0105 warto\u015b\u0107<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Poni\u017cszy przyk\u0142ad pokazuje, jak zastosowa\u0107 t\u0119 formu\u0142\u0119 w praktyce.<\/span><\/p>\n<h2> <span style=\"color: #000000;\"><strong>Przyk\u0142ad: Obliczenie oczekiwanej warto\u015bci X <sup>3<\/sup><\/strong><\/span><\/h2>\n<p> <span style=\"color: #000000;\">Za\u0142\u00f3\u017cmy, \u017ce mamy nast\u0119puj\u0105c\u0105 tabel\u0119 rozk\u0142adu prawdopodobie\u0144stwa, kt\u00f3ra opisuje prawdopodobie\u0144stwo, \u017ce zmienna losowa,<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\" wp-image-31621 aligncenter\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/x2_1.jpg\" alt=\"\" width=\"266\" height=\"253\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Aby obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X <sup>3<\/sup> , mo\u017cemy skorzysta\u0107 z nast\u0119puj\u0105cego wzoru:<\/span><\/p>\n<p> <span style=\"color: #000000;\">E(X <sup>3<\/sup> ) = \u03a3x <sup>3<\/sup> * p(x)<\/span><\/p>\n<p> <span style=\"color: #000000;\">E(X <sup>3<\/sup> ) = (0) <sup>3<\/sup> *.06 + (1) <sup>3<\/sup> *.15 + (2) <sup>3<\/sup> *.17 + (3) <sup>3<\/sup> *.24 + (4) <sup>3<\/sup> *.23 + (5) <sup>3<\/sup> *.09 + (6) <sup>3<\/sup> *.06<\/span><\/p>\n<p> <span style=\"color: #000000;\">E(X <sup>3<\/sup> ) = 0 + 0,15 + 0,1,36 + 6,48 + 14,72 + 11,25 + 12,96<\/span><\/p>\n<p> <span style=\"color: #000000;\">E( <sup>X3<\/sup> ) = 45,596<\/span><\/p>\n<p> <span style=\"color: #000000;\">Oczekiwana warto\u015b\u0107 X <sup>3<\/sup> wynosi <strong>45 596<\/strong> .<\/span><\/p>\n<p> <span style=\"color: #000000;\">Nale\u017cy zauwa\u017cy\u0107, \u017ce ta zmienna losowa jest <strong>dyskretn\u0105 zmienn\u0105 losow\u0105<\/strong> , co oznacza, \u017ce mo\u017ce przyjmowa\u0107 tylko sko\u0144czon\u0105 liczb\u0119 warto\u015bci.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Je\u015bli X jest <strong>ci\u0105g\u0142\u0105 zmienn\u0105 losow\u0105<\/strong> , musimy zastosowa\u0107 nast\u0119puj\u0105cy wz\u00f3r do obliczenia oczekiwanej warto\u015bci X <sup>3<\/sup> :<\/span><\/p>\n<p> <strong><span style=\"color: #000000;\">E(X <sup>3<\/sup> ) = \u222b x <sup>3<\/sup> f(x)dx<\/span><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Z\u0142oto:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">\u222b: Symbol oznaczaj\u0105cy \u201eintegracj\u0119\u201d<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>f(x)<\/strong> : Plik <a href=\"https:\/\/statorials.org\/pl\/cdf-kontra-pdf\/\" target=\"_blank\" rel=\"noopener\">PDF<\/a> jest kontynuowany dla zmiennej losowej<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Przy obliczaniu oczekiwanej <sup>warto\u015bci<\/sup><\/span><\/p>\n<h2> <span style=\"color: #000000;\"><strong>Dodatkowe zasoby<\/strong><\/span><\/h2>\n<p> <span style=\"color: #000000;\">Poni\u017csze samouczki wyja\u015bniaj\u0105, jak wykonywa\u0107 inne typowe zadania w statystykach:<\/span><\/p>\n<p> <a href=\"https:\/\/statorials.org\/pl\/srednia-rozk\u0142adu-prawdopodobienstwa\/\" target=\"_blank\" rel=\"noopener\">Jak znale\u017a\u0107 \u015bredni\u0105 rozk\u0142adu prawdopodobie\u0144stwa<\/a><br \/> <a href=\"https:\/\/statorials.org\/pl\/odchylenie-standardowe-rozk\u0142adu-prawdopodobienstwa\/\" target=\"_blank\" rel=\"noopener\">Jak znale\u017a\u0107 odchylenie standardowe rozk\u0142adu prawdopodobie\u0144stwa<\/a><br \/> <a href=\"https:\/\/statorials.org\/pl\/wariancja-rozk\u0142adu-prawdopodobienstwa\/\" target=\"_blank\" rel=\"noopener\">Jak znale\u017a\u0107 wariancj\u0119 rozk\u0142adu prawdopodobie\u0144stwa<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>W przypadku zmiennej losowej oznaczonej X mo\u017cna zastosowa\u0107 nast\u0119puj\u0105cy wz\u00f3r do obliczenia oczekiwanej warto\u015bci X 3 : E(X 3 ) = \u03a3x 3 * p(x) Z\u0142oto: \u03a3 : Symbol oznaczaj\u0105cy \u201esum\u0119\u201d x : Warto\u015b\u0107 zmiennej losowej p(x) : Prawdopodobie\u0144stwo, \u017ce zmienna losowa przyjmie dan\u0105 warto\u015b\u0107 Poni\u017cszy przyk\u0142ad pokazuje, jak zastosowa\u0107 t\u0119 formu\u0142\u0119 w praktyce. Przyk\u0142ad: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-3935","post","type-post","status-publish","format-standard","hentry","category-przewodnik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3 - Statologia<\/title>\n<meta name=\"description\" content=\"W tym samouczku wyja\u015bniono, jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3, na przyk\u0142adzie.\" \/>\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\/wartosc-oczekiwana-x3\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3 - Statologia\" \/>\n<meta property=\"og:description\" content=\"W tym samouczku wyja\u015bniono, jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3, na przyk\u0142adzie.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-14T16:01:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/x2_1.jpg\" \/>\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=\"1 minuta\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/\",\"url\":\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/\",\"name\":\"Jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3 - Statologia\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-07-14T16:01:08+00:00\",\"dateModified\":\"2023-07-14T16:01:08+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"W tym samouczku wyja\u015bniono, jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 X^3, na przyk\u0142adzie.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/wartosc-oczekiwana-x3\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Jak obliczy\u0107 oczekiwan\u0105 warto\u015b\u0107 x^3\"}]},{\"@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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