{"id":21999,"date":"2016-11-30T12:16:31","date_gmt":"2016-11-30T17:16:31","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=1049"},"modified":"2026-04-04T11:28:04","modified_gmt":"2026-04-04T16:28:04","slug":"variables-aleatorias-discretas-tabla","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-recursos\/variables-aleatorias-discretas-tabla\/","title":{"rendered":"Variables Aleatorias Discretas"},"content":{"rendered":"\n

Referencia<\/strong>: Le\u00f3n-Garc\u00eda 3.5 Important Discrete Random Variables p115 <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Bernoulli<\/h3>\n\n\n\n
\n\n\n S_X=\\{0, 1 \\} <\/span>\n\n\n p_0 = q = 1-p <\/span>\n\n\n p_1=p, 0 \\leq p \\leq 1 <\/span>\n\n\n E[X] = p <\/span>\n\n\n VAR[x] = p(1-p) <\/span>\n\n\n G_X(z)=(q+pz) <\/span>\n\n\n\n

Nota: La variable aleatoria Bernoulli es es valor de la funci\u00f3n indicador IA<\/sub> para alg\u00fan evento; X=1 si ocurre A, y 0 en otro caso <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Binomial<\/h3>\n\n\n\n
\n\n\n S_X=\\{0, 1, ... , n \\} <\/span>\n\n\n p_k={n \\choose k} p^{k} (1-p)^{n-k} <\/span>\n\n\n k = 0, 1, ... , n <\/span>\n\n\n E[X]= np <\/span>\n\n\n VAR[X] = np(1-p) <\/span>\n\n\n G_X(z)= (q + pz)^{n} <\/span>\n\n\n\n

Nota: X es el numero de \u00e9xtidos en n intentos Bernoulli y por consiguiente la suma de n iid variables aleatorias Bernoulli. <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Geom\u00e9trica<\/h3>\n\n\n\n
\n\n\n\n

Versi\u00f3n 1:<\/p>\n\n\n S_X=\\{0, 1, 2, ... \\} <\/span>\n\n\n p_k = p(1-p)^{k} <\/span>\n\n\n k= 0, 1, , ... <\/span>\n\n\n E[X] =\\frac{1-p}{p}<\/span>\n\n\n VAR[X] = \\frac{1-p}{p^2}<\/span>\n\n\n G_X(z) = \\frac{p}{1-qz} <\/span>\n\n\n\n

Nota: X es el n\u00famero de fallas antes del primer \u00e9xito en una secuencia de intentos Bernoulli independientes. La variable aleatoria geom\u00e9trica es la \u00fanica una variable aleatoria con propiedad \"sin memoria\". <\/em><\/p>\n\n\n\n

Versi\u00f3n 2:<\/p>\n\n\n S_X'=\\{ 1, 2, ... \\} <\/span>\n\n\n p_k = p(1-p)^{k-1} <\/span>\n\n\n k= 1, 2, ... <\/span>\n\n\n E[X'] =\\frac{1}{p}<\/span>\n\n\n VAR[X'] = \\frac{1-p}{p^2}<\/span>\n\n\n G_{X'}(z) = \\frac{pz}{1-qz} <\/span>\n\n\n\n

Nota: X'= X+1 es el n\u00famero de intentos hasta primer \u00e9xito en una secuencia de intentos Bernoulli independientes. <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Binomial Negativa<\/h3>\n\n\n\n
\n\n\n S_X=\\{ r, r+1, ... \\}<\/span>\n\n\n\n

, donde r es un entero positivo<\/p>\n\n\n p_k = {{k-1} \\choose {r-1}} p^{r}(1-p)^{k-r} <\/span>\n\n\n k = r, r+1, ... <\/span>\n\n\n E[x] = \\frac{r}{p}<\/span>\n\n\n VAR[X] = \\frac{r(1-p)}{p^2}<\/span>\n\n\n G_X(z) = \\left( \\frac{pz}{1-qz}\\right)^{r} <\/span>\n\n\n\n

Nota: X es el n\u00famero de intentos hasta el r-\u00e9simo \u00e9xito en una secuencia de intentos Bernoulli independientes <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Poisson<\/h3>\n\n\n\n
\n\n\n S_X=\\{0, 1, 2, ... \\} <\/span>\n\n\n p_k = \\frac{\\alpha ^{k}}{k!} e^{-\\alpha} <\/span>\n\n\n k = 0, 1, ... \\text{ y } \\alpha>0<\/span>\n\n\n E[X] = \\alpha<\/span>\n\n\n VAR[X] = \\alpha<\/span>\n\n\n G_X(z) = e^{\\alpha(z-1)}<\/span>\n\n\n\n

Nota: X es el n\u00famero de eventos que ocurren en una unidad de tiempo cuando el tiempo entre eventos es distribuido exponencialmente con media 1\/\u03b1. <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Uniforme<\/h3>\n\n\n\n
\n\n\n S_X=\\{1, 2, ..., L \\}<\/span>\n\n\n p_k = \\frac{1}{L}<\/span>\n\n\n k = 1, 2, ... , L <\/span>\n\n\n E[X] = \\frac{L+1}{2}<\/span>\n\n\n VAR[X] = \\frac{L^2 -1}{12}<\/span>\n\n\n G_X(z) = \\frac{z}{L} \\frac{1- z^L}{1-z}<\/span>\n\n\n\n

Nota: La variable aleatoria uniforme sus resultados son igualmente probables. Juega un rol importante en la generaci\u00f3n de n\u00fameros aleatorios <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Zipf<\/h3>\n\n\n\n
\n\n\n S_X=\\{1, 2, ..., L \\} <\/span>\n\n\n\n

, donde L es un entero positivo<\/p>\n\n\n p_k = \\frac{1}{c_L} \\frac{1}{k} <\/span>\n\n\n k = 1, 2, ... , L <\/span>\n\n\n\n

donde cL<\/sub> esta dado por:<\/p>\n\n\n c_L = \\sum_{j=1}^{L} = 1 + \\frac{1}{2} + \\frac{1}{3} + ... + \\frac{1}{L} <\/span>\n\n\n E[X] =\\frac{L}{c_L}<\/span>\n\n\n VAR[X] = \\frac{L(L+1)}{2C_L} - \\frac{L^2}{c_L ^2}<\/span>\n\n\n\n

Nota: La variable aleatoria Zipf tiene la propiedad que algunos resultados ocurren frecuentemente, pero muchos resultados suceden muy poco <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

Bernoulli<\/a><\/p>\n\n\n\n

Binomial<\/a><\/p>\n\n\n\n

Geom\u00e9trica<\/a><\/p>\n\n\n\n

Binomial Negativa<\/a><\/p>\n\n\n\n

Poisson<\/a><\/p>\n\n\n\n

Uniforme<\/a><\/p>\n\n\n\n

Zipf<\/a><\/p>\n<\/div>\n\n\n\n

\n","protected":false},"excerpt":{"rendered":"

Referencia: Le\u00f3n-Garc\u00eda 3.5 Important Discrete Random Variables p115 Bernoulli Binomial Geom\u00e9trica Binomial Negativa Poisson Uniforme Zipf Bernoulli Nota: La variable aleatoria Bernoulli es es valor de la funci\u00f3n indicador IA para alg\u00fan evento; X=1 si ocurre A, y 0 en otro caso Bernoulli Binomial Geom\u00e9trica Binomial Negativa Poisson Uniforme Zipf Binomial Nota: X es el […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-stp-unidades","format":"standard","meta":{"footnotes":""},"categories":[205],"tags":[],"class_list":["post-21999","post","type-post","status-publish","format-standard","hentry","category-stp-recursos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/21999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=21999"}],"version-history":[{"count":4,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/21999\/revisions"}],"predecessor-version":[{"id":22288,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/21999\/revisions\/22288"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=21999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=21999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=21999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}