{"id":497,"date":"2016-11-11T06:48:28","date_gmt":"2016-11-11T11:48:28","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=497"},"modified":"2026-04-04T10:49:07","modified_gmt":"2026-04-04T15:49:07","slug":"valor-esperado-funcion-variable-aleatoria","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-u01eva\/valor-esperado-funcion-variable-aleatoria\/","title":{"rendered":"Valor Esperado de una funci\u00f3n de variable aleatoria"},"content":{"rendered":"\n<p><em><strong>Referencia<\/strong><\/em>: Gubner 2.4 p83 , Ross 2.4.3 p42, Le\u00f3n-Garc\u00eda 3.3.1 p.107<\/p>\n\n\n\n<p>Dada una variable aleatoria X, se puede definir una nueva variable aleatoria Z = g(X), donde g(x) es una funci\u00f3n de valor real de la variable real x.<\/p>\n\n\n\n<p>Para calcular E[Z] se puede proceder como:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> E[g(X)] = \\sum_i g(x_i) p_X(x_i) <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> E[g(x)] = \\int_{-\\infty}^{\\infty} g(x) f(x) dx <\/span>\n\n\n\n<p>dado que la f\u00f3rmula es mas f\u00e1cil de usar que encontrar la pmf de Z, la formula se la conoce como la \"ley del estad\u00edstico inconsciente\" o LOTUS (Law Of The Unconscious Statistician).<\/p>\n\n\n\n<p>Una aplicaci\u00f3n simple es :<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> E[aX] = \\sum_i ax_i p_X(x_i) = <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> = a \\sum_i x_i p_X(x_i) = a E[X] <\/span>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h3 class=\"wp-block-heading\">Ejemplo<\/h3>\n\n\n\n<p><em><strong>Referencia<\/strong><\/em>: Le\u00f3n- Garc\u00eda 3.17 p107<\/p>\n\n\n\n<p>Sea X el ruido en el voltaje que est\u00e1 uniformemente distribuido en S<sub>X<\/sub> = {-3,-1,+1,+3} con p<sub>X<\/sub> (k) =&nbsp; 1\/4 para k en S<sub>X<\/sub>. Encuentre E[Z] donde Z=X<sup>2<\/sup>.<\/p>\n\n\n\n<p><em>Soluci\u00f3n<\/em>: Se busca primero encontrar la pmf (probability mass function) de Z, el S<sub>Z<\/sub> ={9,1,1,9} = {1,9}, por lo que:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>p<sub>Z<\/sub>(9) = P&#091;X \u2208 {-3,+3}] \n      = p<sub>X<\/sub>(-3) + p<sub>X<\/sub>(3) \n      = 1\/4 + 1\/4 = 1\/2\np<sub>Z<\/sub>(1) = p<sub>X<\/sub>(-1) + p<sub>X<\/sub>(1) = \n      = 1\/4 + 1\/4 = 1\/2\nentonces:\nE&#091;Z] = 1(1\/2) + 9(1\/2) = 5 \n<\/code><\/pre>\n\n\n\n<p>usando la f\u00f3rmula para E[Z]:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> E[Z] = E[g(X)] = \\sum_i g(x_i)p_X(x_i) = <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\sum_i i^2 p_X(x_i) = \\frac{1}{4} [(-3)^2 + (-1)^2+1^2+2^2] =<\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> = \\frac{20}{4} = 5 <\/span>\n\n\n\n<p>con lo que se obtuvo el mismo resultado.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p>Ross Corolario 2.2. Siendo a y b constantes, entonces:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> E[aX + b] = aE[X] +b <\/span>\n","protected":false},"excerpt":{"rendered":"<p>Referencia: Gubner 2.4 p83 , Ross 2.4.3 p42, Le\u00f3n-Garc\u00eda 3.3.1 p.107 Dada una variable aleatoria X, se puede definir una nueva variable aleatoria Z = g(X), donde g(x) es una funci\u00f3n de valor real de la variable real x. Para calcular E[Z] se puede proceder como: dado que la f\u00f3rmula es mas f\u00e1cil de usar [&hellip;]<\/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":[213],"tags":[],"class_list":["post-497","post","type-post","status-publish","format-standard","hentry","category-stp-u01eva"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/497","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=497"}],"version-history":[{"count":1,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/497\/revisions"}],"predecessor-version":[{"id":22113,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/497\/revisions\/22113"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=497"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=497"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}