{"id":34184,"date":"2021-06-13T13:00:53","date_gmt":"2021-06-13T13:00:53","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34184"},"modified":"2025-08-16T11:16:37","modified_gmt":"2025-08-16T11:16:37","slug":"variabiles-casuales-et-distributiones-probabilitatum","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/","title":{"rendered":"Variabiles Casuales et Distributiones Probabilitatum"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Variabiles Casuales et Distributiones Probabilitatum<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summarium<\/strong><br \/><em>Haec lectio profundam immersionem praebet in notiones variabilium casualium et distributionum probabilitatis, columnas fundamentales theoriae probabilitatum atque analysis statisticae. Introducitur definitio variabilis casualis ut numerus qui a exitu experimenti casus pendet. Tractatur functio distributionis variabilis casualis, eius momentum illustrando, necnon proprietates eius essentiales. Denique perscrutatur relatio inter variabiles casuales et distributiones probabilitatis, explicando duas variabiles eandem distributionem habere posse sine eo quod eadem variabilis casualis sint.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nCum haec lectio perfecta erit, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> notionem variabilium casualium: Discipuli capaces esse debent describere atque explicare quid sint variabiles casuales et quomodo mathematice definiantur.<\/li>\n<li><strong>Intellegere<\/strong> notionem distributionum probabilitatis: Discipuli explicare debent quid sint distributiones probabilitatis et quomodo repraesententur.<\/li>\n<li><strong>Describere<\/strong> proprietates distributionum probabilitatis: Discipuli agnoscere atque explicare debent proprietates claves distributionum probabilitatis.<\/li>\n<li><strong>Analyzare<\/strong> relationem inter variabiles casuales et distributiones probabilitatis: Discipuli disputare debent quomodo variabiles casuales et distributiones probabilitatis inter se coniungantur, et quomodo duae variabiles eandem distributionem habere possint sine eo quod eadem variabilis casualis sint.<\/li>\n<li><strong>Demonstrate<\/strong> et applicare proprietates distributionum probabilitatis in condicionibus practicis: Discipuli proprietates distributionum probabilitatis mathematice demonstrare et eas in rebus realibus applicare debent.<\/li>\n<li><strong>Intellegere<\/strong> notionem functionum distributionis: Discipuli describere debent quid sit functio distributionis et quomodo adhibita sit ad variabilem casualem describendam.<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>Quid sunt variabiles casuales?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Quid sunt distributiones probabilitatum?<\/strong><\/a><br \/>\n<a href=\"#3\">Proprietates distributionum probabilitatis<\/a><br \/>\n<a href=\"#4\">Relatio inter variabiles casuales et distributiones probabilitatum<\/a><br \/>\n<\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/gIKn9t1hnrw\" 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:\">Una ex notionibus clavibus theoriae probabilitatum et analysis statisticae sunt variabiles casuales et distributiones probabilitatum. Quamvis theoria quam adhuc evolvimus quodammodo \u00abcompleta\u00bb sit, veritas est quod in statu hodierno satis rudis est; variabiles casuales et distributiones probabilitatis sunt, ut ita dicamus, notiones quae nobis permittunt \u00abungere facultatem nostram ad operandum cum probabilitatibus et faciendum analysin statisticam\u00bb.<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>Quid sunt variabiles casuales?<\/h2>\n<p style=\"text-align: justify; color:\"><strong><span style=\"color: #ff0000;\">Ut nos cum notione variabilis casualis assuescamus<\/span><\/strong>, utile est initium sumere ab accessu intuitivo: variabilis casualis interpretari potest ut \u00abnumerus qui a exitu experimenti casus pendet\u00bb. Nihilominus, ad intellegentiam accuratiorem, essentiale est etiam definitionem formalem explorare. Videamus hanc definitionem:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color:\"><span style=\"color: #000080;\"><strong>Definitio:<\/strong><\/span> Variabilis casualis super coniunctione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> est functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color:\">Casus communissimus est cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> et, nisi aliter specificetur, hoc deinceps assumemus; id est, operabimur cum variabilibus casualibus valoribus realibus. Generaliter, variabiles casuales litteris maiusculis denotantur, ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span>, dum constantes litteris minusculis denotantur. Ad simpliciorem usum, variabiles casuales simpliciter \u00abvariabiles\u00bb appellabimus.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color:\"><span style=\"color:\"><strong>Exemplum:<\/strong><\/span> Ponamus datum sex facierum bis iactari. Tunc habebimus:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2d6} = \\{(\\omega_1, \\omega_2)\\;|\\; \\omega_1,\\omega_2 \\in \\{1,2,3,4,5,6\\}\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Ex hoc definire possumus sequentes variabiles casuales:<\/p>\n<ul style=\"text-align: justify; color:\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u00abNumerus vicium quibus unus apparet\u00bb<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u00abSumma eventuum obtentorum\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\omega_1 + \\omega_2<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z=<\/span><\/span> \u00abEventus secundi iactus\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\omega_2<\/span><\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>Quid sunt distributiones probabilitatum?<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color:\"><span style=\"color: #000080;\"><strong>Definitio: <\/strong><\/span><strong><span style=\"color: #ff0000;\">Functio distributionis (vel \u00abFD\u00bb)<\/span><\/strong> variabilis casualis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> est functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span> definita per relationem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> vel breviori modo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\leq x).<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color:\">Generatim, quod interest de variabili casuali non est tam expressio eius explicita in spatio exemplari <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span>, sed functio eius distributionis. Subscriptum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> omitti potest si contextus clarus est et nulla ambiguitas exsistit. Commune est uti notatione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> ad significandum variabilem casualem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> functionem distributionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> habere.<\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>Proprietates distributionum probabilitatis<\/h3>\n<p style=\"text-align: justify; color:\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> est distributio probabilitatum<\/span><\/strong><\/a> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> sunt numeri reales quicumque, tunc sequentia proprietates valebunt:<\/p>\n<p style=\"text-align: justify; color:\">(a) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow [P(a\\lt X \\leq b) = F(b) - F(a)]<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> id est, \u00abF est crescens\u00bb.<\/p>\n<p style=\"text-align: justify; color:\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to -\\infty} F(x) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(d) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=\\lim_{t\\to x^+}F(t) - \\lim_{t\\to x^-}F(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">(e) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F(x)=\\lim_{t\\to x^+}F(t)<\/span><\/span><\/p>\n<table style=\"text-align: justify; color:\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e4316b3a991\"  tabindex=\"0\" title=\"DEMONSTRATIO\"    >DEMONSTRATIO<\/span><div id=\"target-id69e4316b3a991\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color:\"><strong>(a)<\/strong> Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> eventus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq a\\}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq b\\}<\/span><\/span> respective, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b.<\/span><\/span> Si hoc totum accidit, tum erit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span> atque ideo fiet ut<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(a\\lt X\\leq b)} = P(B\\setminus A) = P(B) - P(B\\cap A) = P(B)-P(A) =\\color{blue}{F(b) - F(a)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(b)<\/strong> Ex parte (a) habetur quod: Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0,<\/span><\/span> tunc habetur:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(b) - F(a) \\geq 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">quod idem est ac dicere<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a) \\leq F(b)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(c)<\/strong> Hic utemur eo quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> est monotona crescens (in (b) probatum) et limitata cum valore maximo aequali \u00ab1\u00bb (quia distributio definitur in terminis probabilitatis). Hoc solum sufficit ad dicendum quod<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Accessus complementarius ad hoc nobis permittit sequentia computare cum eodem effectu.<\/p>\n<p style=\"text-align: justify; color:\">Definiamus coniunctionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> Ex hoc facile est verificare quod, pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> eveniet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span> atque ideo, utens proprietate <a href=\"http:\/\/toposuranos.com\/material\/es\/teoremas-utiles-para-el-calculo-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">continuitatis<\/a> habebitur:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1=P(\\Omega) = P\\left( \\bigcup_{n\\lt +\\infty} A_n \\right) = \\lim_{n\\to +\\infty} P(A_n) = \\lim_{n\\to +\\infty} P(\\{\\omega\\;|\\;X(\\omega)\\leq n\\}) = \\lim_{n\\to +\\infty} P(X\\leq n)=\\lim_{n\\to +\\infty}F(n)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Id est:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\lim_{x\\to +\\infty} F(x) = 1}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Contra, pro limite in quo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span>, habetur sequens:<\/p>\n<p style=\"text-align: justify; color:\">Primum definiamus coniunctionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> Ex hoc verificatur quod:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n \\to -\\infty}F(n) = \\lim_{n\\to -\\infty} P(X\\leq n) = \\lim_{n\\to \\infty} P(X\\leq -n)= 1 - \\lim_{n\\to \\infty} P(-n \\lt X) = 1 - \\lim_{n\\to \\infty}P(B_n)) = 1 - P(\\Omega) = 1-1=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(d)<\/strong> Similiter ac in parte (c) ratiocinatur. Incipitur definendo coniunctionem<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_n = \\left\\{x - \\frac{1}{n} \\leq X \\leq x + \\frac{1}{n}\\right\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Et ex hoc habetur quod<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> C_{n+1}\\subseteq C_n<\/span><\/span><\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcap_{n\\gt 0} C_n = \\{X=x\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Itaque, utens <a href=\"http:\/\/toposuranos.com\/material\/es\/teoremas-utiles-para-el-calculo-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">consecutione ex proprietate continuitatis<\/a> habetur:<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=P\\left(\\bigcap_{n\\gt 0} C_n \\right) = \\lim_{n\\to \\infty} P(C_n) = \\lim_{x+1\/n \\to x^+}F\\left(x+1\/n\\right) - \\lim_{x-1\/n \\to x^-}F\\left(x-1\/n\\right)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\"><strong>(e)<\/strong> Hic ultimus casus ex antecedenti effectu obtinetur. Revera, cum iam probaverimus<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Possumus scribere<\/p>\n<p style=\"text-align: center; color:\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{t \\to x^+}F\\left(t\\right) = P(X=x) + \\lim_{t \\to x^-}F\\left(t\\right) = P(X=x) + \\lim_{t\\to x^-}P(X\\leq t)= P(X\\leq x) = F(x)<\/span><\/span><\/p>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>Relatio inter variabiles casuales et distributiones probabilitatum<\/h3>\n<p style=\"text-align: justify; color:\">Dicitur duas variabiles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> eandem distributionem probabilitatis habere si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall A\\subseteq \\mathbb{R})(P(X\\in A) = P(Y\\in A)).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color:\">Duae variabiles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> definitae super idem spatium exemplare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> eandem distributionem habere possunt, sed non ideo sunt necessario eadem variabilis casualis. Exempli gratia, si consideremus experimentum iactandi nummum aequilibratum duarum facierum et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> respondet capiti et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> respondet nummi basi, definiri potest variabilis casualis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span> et habebitur quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5,<\/span><\/span> atque ambae eandem distributionem habent, sed si computetur probabilitas ut ambae eundem valorem habeant habebitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Variabiles Casuales et Distributiones Probabilitatum SummariumHaec lectio profundam immersionem praebet in notiones variabilium casualium et distributionum probabilitatis, columnas fundamentales theoriae probabilitatum atque analysis statisticae. Introducitur definitio variabilis casualis ut numerus qui a exitu experimenti casus pendet. Tractatur functio distributionis variabilis casualis, eius momentum illustrando, necnon proprietates eius essentiales. Denique perscrutatur relatio inter variabiles casuales et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26787,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1298,1364],"tags":[],"class_list":["post-34184","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-probabilitates-et-statistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Variabiles Casuales et Distributiones Probabilitatum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.\" \/>\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\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Variabiles Casuales et Distributiones Probabilitatum\" \/>\n<meta property=\"og:description\" content=\"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\" \/>\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=\"2021-06-13T13:00:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-16T11:16:37+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Variabiles Casuales et Distributiones Probabilitatum\" \/>\n<meta name=\"twitter:description\" content=\"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Variabiles Casuales et Distributiones Probabilitatum\",\"datePublished\":\"2021-06-13T13:00:53+00:00\",\"dateModified\":\"2025-08-16T11:16:37+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\"},\"wordCount\":1341,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\",\"articleSection\":[\"Mathematica\",\"Probabilitates et Statistica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\",\"name\":\"Variabiles Casuales et Distributiones Probabilitatum - toposuranos.com\/material\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\",\"datePublished\":\"2021-06-13T13:00:53+00:00\",\"dateModified\":\"2025-08-16T11:16:37+00:00\",\"description\":\"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.\",\"breadcrumb\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\",\"width\":1024,\"height\":356,\"caption\":\"Created with GIMP\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Variabiles Casuales et Distributiones Probabilitatum\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\/\/toposuranos.com\/material\/#website\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"name\":\"toposuranos.com\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\/\/toposuranos.com\/material\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\",\"name\":\"toposuranos.com\/material\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\/material\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/groups\/toposuranos\",\"https:\/\/x.com\/topuranos\",\"https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\/\/www.linkedin.com\/company\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\/\/toposuranos.com\/material\"],\"url\":\"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Variabiles Casuales et Distributiones Probabilitatum - toposuranos.com\/material","description":"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/","og_locale":"es_ES","og_type":"article","og_title":"Variabiles Casuales et Distributiones Probabilitatum","og_description":"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.","og_url":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-06-13T13:00:53+00:00","article_modified_time":"2025-08-16T11:16:37+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Variabiles Casuales et Distributiones Probabilitatum","twitter_description":"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Variabiles Casuales et Distributiones Probabilitatum","datePublished":"2021-06-13T13:00:53+00:00","dateModified":"2025-08-16T11:16:37+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/"},"wordCount":1341,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","articleSection":["Mathematica","Probabilitates et Statistica"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/","url":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/","name":"Variabiles Casuales et Distributiones Probabilitatum - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","datePublished":"2021-06-13T13:00:53+00:00","dateModified":"2025-08-16T11:16:37+00:00","description":"Discere notiones essentiales variabilium casualium et distributionum probabilitatis in hac lectione diligenter exposita.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg","width":1024,"height":356,"caption":"Created with GIMP"},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/la\/variabiles-casuales-et-distributiones-probabilitatum\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Variabiles Casuales et Distributiones Probabilitatum"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/34184","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=34184"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/34184\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/26787"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=34184"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=34184"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=34184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}