{"id":1491,"date":"2016-11-20T08:54:47","date_gmt":"2016-11-20T13:54:47","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=1491"},"modified":"2026-04-04T10:50:02","modified_gmt":"2026-04-04T15:50:02","slug":"media-y-autocovarianza-con-t","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-u01eva\/media-y-autocovarianza-con-t\/","title":{"rendered":"Media y autocovarianza con t"},"content":{"rendered":"

Referencia<\/em><\/strong>: Problema 9.13 Leon-Garc\u00eda p.558 pdf 55<\/p>\n

El proceso aleatorio Z(t) definido por:<\/p>\n

Z<\/strong>(t) = 2X<\/strong>t -Y<\/strong><\/p>\n

donde X y Y son\u00a0 variables aleatorias con medias mX<\/sub>, mY<\/sub> ,\u00a0 varianzas \u03c32<\/sup>X<\/sub> y \u03c32<\/sup>Y<\/sub> y coeficientes de correlaci\u00f3n \u03c1XY<\/sub>.<\/p>\n

Encuentre la media y autocovarianza de Z(t)<\/p>\n

\n

Soluci\u00f3n propuesta<\/strong><\/em>:<\/p>\n

E[Z(t)] = E[ 2Xt - Y ]\n    = 2E[X]t - E[Y]\n    = 2tmX<\/sub> - mY<\/sub> = mZ<\/sub><\/pre>\n
CZ<\/sub>(t1<\/sub>, t2<\/sub>) = E[(2Xt1<\/sub> - Y)(2Xt2<\/sub> - Y)] - mZ<\/sub>(t1<\/sub>) mZ<\/sub>(t2<\/sub>)\n    = E[ 4X2<\/sup> t1<\/sub>t2<\/sub> - 2XYt1<\/sub>   - 2XYt2<\/sub> + Y2<\/sup>] \n      - (2t1<\/sub>mX<\/sub> - mY<\/sub>)(2t2<\/sub>mX<\/sub> - mY<\/sub>)\n    = 4t1<\/sub>t2<\/sub> E[X2<\/sup>] - 2t1<\/sub> E[XY] - 2t2<\/sub> E[XY] + E[Y2<\/sup>] \n      - (4t1<\/sub>t2<\/sub>m2<\/sup>X<\/sub> -2t1<\/sub>mX<\/sub>mY<\/sub> -2t2<\/sub>mX<\/sub>mY<\/sub> + m2<\/sup>Y<\/sub>)\n    = 4t1<\/sub>t2<\/sub> E[X2<\/sup>] - 2(t1<\/sub> +t2<\/sub>)E[XY] + E[Y2<\/sup>] \n      -4t1<\/sub>t2<\/sub>m2<\/sup>X<\/sub> + 2(t1<\/sub>+2<\/sub>)mX<\/sub>mY<\/sub> - m2<\/sup>Y<\/sub> \n   \u00a0= 4t1<\/sub>t2<\/sub> (E[X2<\/sup>] - m2<\/sup>X<\/sub> ) - 2(t1<\/sub> +t2<\/sub>)(E[XY] - mX<\/sub>mY<\/sub>) + (E[Y2<\/sup>] - m2<\/sup>Y<\/sub>)\n    = 4t1<\/sub>t2<\/sub> \u03c32<\/sup>X<\/sub> - 2(t1<\/sub> +t2<\/sub>)\u03c3XY<\/sub> + \u03c32<\/sup>Y<\/sub> \n\ndado \u03c1XY<\/sub> = \u03c3XY<\/sub>\/\u03c3X<\/sub>\u03c3Y<\/sub>\n     \u03c1XY<\/sub>\u03c3X<\/sub>\u03c3Y<\/sub> = \u03c3XY<\/sub>\n\n    = 4t1<\/sub>t2<\/sub> \u03c32<\/sup>X<\/sub> - 2(t1<\/sub> +t2<\/sub>) \u03c1XY<\/sub>\u03c3X<\/sub>\u03c3Y<\/sub>  + \u03c32<\/sup>Y<\/sub> \n\n\u03c32<\/sup>Z(t)<\/sub> = CZ<\/sub>(t, t) = 4t2<\/sup>\u03c32<\/sup>X<\/sub> - 4t \u03c1XY<\/sub>\u03c3X<\/sub>\u03c3Y<\/sub> + \u03c32<\/sup>Y<\/sub> \n<\/pre>\n
Para X y Y solo se da la media y varianza, la funci\u00f3n final debe tener la misma forma:
\n f_{Z(t)} = \\frac{e^{ -(z - m_Z)^2 \/(2 \\sigma_Z^2)}}{ \\sqrt{2 \\pi }\\sigma_Z} <\/span><\/p>\n f_{Z(t)} = \\frac{e^{-\\frac{(z - 2t m_{X} +m_{Y})^2 }{(2(4t^2\\sigma_{X}^2 - 4t\\sigma_{X} \\sigma_{Y} \\rho_{XY} + \\sigma_{Y}^2)}}}{ \\sqrt{2 \\pi }\\sqrt{4t^2\\sigma_{X}^2 - 4t\\sigma_{X} \\sigma_{Y} \\rho_{XY} + \\sigma_{Y}^2}} <\/span>\n","protected":false},"excerpt":{"rendered":"
Referencia: Problema 9.13 Leon-Garc\u00eda p.558 pdf 55 El proceso aleatorio Z(t) definido por: Z(t) = 2Xt -Y donde X y Y son\u00a0 variables aleatorias con medias mX, mY ,\u00a0 varianzas \u03c32X y \u03c32Y y coeficientes de correlaci\u00f3n \u03c1XY. Encuentre la media y autocovarianza de Z(t) Soluci\u00f3n propuesta: E[Z(t)] = E[ 2Xt - Y ] = […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-stp-ejercicios","format":"standard","meta":{"footnotes":""},"categories":[213],"tags":[],"class_list":["post-1491","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\/1491","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=1491"}],"version-history":[{"count":1,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1491\/revisions"}],"predecessor-version":[{"id":22263,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1491\/revisions\/22263"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=1491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=1491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=1491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}