{"id":1697,"date":"2016-10-08T18:35:27","date_gmt":"2016-10-08T23:35:27","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=1697"},"modified":"2026-04-05T16:44:55","modified_gmt":"2026-04-05T21:44:55","slug":"s2eva2017tii_t2-covarianza-xy","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-ejemplos\/s2eva2017tii_t2-covarianza-xy\/","title":{"rendered":"s2Eva2017TII_T2 Covarianza X, Y"},"content":{"rendered":"\n

Ejercicio<\/strong>: 2Eva2017TII_T2 Covarianza X, Y<\/a><\/p>\n\n\n\n

Tema 2<\/strong>.<\/p>\n\n\n\n

\u03b8<\/strong> es una variable aleatoria uniforme, distribuida en el rango [-\u03c0 \u03c0].<\/p>\n\n\n f_{\\theta} (\\theta) = \\frac{1}{\\pi - (-\\pi))} = \\frac{1}{2\\pi} <\/span>\n\n\n\n

valor esperados X(t)<\/p>\n\n\n X(t) = \\cos (\\omega t + \\theta) <\/span>\n\n\n E[X(t)] = E[ \\cos (\\omega t + \\theta)] <\/span>\n\n\n = \\int_{-\\pi}^{\\pi}x(t)f_{\\theta}(\\theta) d\\theta <\/span>\n\n\n = \\int_{-\\pi}^{\\pi}\\cos (\\omega t + \\theta) \\frac{1}{2\\pi} d\\theta <\/span>\n\n\n = \\frac{1}{2\\pi}\\sin (\\omega t + \\theta) \\Big|_{-\\pi}^{\\pi} <\/span>\n\n\n = \\frac{1}{2\\pi} [ \\sin (\\omega t +\\pi) -\\sin (\\omega t -\\pi) ] <\/span>\n\n\n E[X(t)] = 0<\/span>\n\n\n\n

valor esperado Y(t)<\/p>\n\n\n Y(t) = \\sin (\\omega t + \\theta) <\/span>\n\n\n E[Y(t)] = E[ \\sin(\\omega t + \\theta)] <\/span>\n\n\n = \\int_{-\\pi}^{\\pi}y(t)f_{\\theta}(\\theta) d\\theta <\/span>\n\n\n = \\int_{-\\pi}^{\\pi}\\sin(\\omega t + \\theta) \\frac{1}{2\\pi} d\\theta <\/span>\n\n\n = \\frac{1}{2\\pi} [-\\cos(\\omega t + \\theta)] \\Big|_{-\\pi}^{\\pi} <\/span>\n\n\n = \\frac{1}{2\\pi} [ -\\cos(\\omega t +\\pi) - (-\\cos (\\omega t -\\pi)) ] <\/span>\n\n\n = \\frac{1}{2\\pi} [\\cos(\\omega t -\\pi) - \\cos (\\omega t +\\pi) ] <\/span>\n\n\n E[Y(t)] = 0 <\/span>\n\n\n\n

\n\n\n\n

Correlaci\u00f3n X(t) y Y(t)<\/p>\n\n\n R_{XY}[t,t+\\tau] =E[X(t) Y(t+\\tau)] <\/span>\n\n\n =E[\\cos (\\omega t + \\theta) \\sin (\\omega (t+\\tau) + \\theta)] <\/span>\n\n\n =E \\Big[ \\frac{1}{2}\\Big[\\sin [(\\omega (t+\\tau) + \\theta) - (\\omega t + \\theta)] + \\sin [(\\omega (t+\\tau) + \\theta) + (\\omega t + \\theta)] \\Big] \\Big] <\/span>\n\n\n =\\frac{1}{2}E\\Big[\\sin (\\omega \\tau) + \\sin (2\\omega t+ \\omega \\tau + 2\\theta) \\Big] <\/span>\n\n\n =\\frac{1}{2}E \\Big[ \\sin (\\omega \\tau) \\Big] + \\frac{1}{2}E\\Big[\\sin (2\\omega t+ \\omega \\tau + 2\\theta) \\Big] <\/span>\n\n\n\n

El primer t\u00e9rmino no contiene la variable aleatorioa \u0398, por lo que se comporta como una constante para el valor esperado.<\/p>\n\n\n =\\frac{\\sin (\\omega \\tau)}{2} + \\frac{1}{2}\\int_{-\\pi}^{\\pi}\\sin (2\\omega t+ \\omega \\tau + 2\\theta) \\frac{1}{2\\pi} d\\theta <\/span>\n\n\n =\\frac{\\sin (\\omega \\tau)}{2} - \\frac{1}{4\\pi}\\cos (2\\omega t+ \\omega \\tau + 2\\theta) \\Big|_{-\\pi}^{\\pi} <\/span>\n\n\n =\\frac{\\sin (\\omega \\tau)}{2} - \\frac{1}{4\\pi}\\Big[ \\cos (2\\omega t+ \\omega \\tau + 2\\pi) - \\cos (2\\omega t+ \\omega \\tau - 2\\pi)\\Big]<\/span>\n\n\n =\\frac{\\sin (\\omega \\tau)}{2} - 0<\/span>\n\n\n R_{XY}[t,t+\\tau] =\\frac{\\sin (\\omega \\tau)}{2} <\/span>\n\n\n\n

\n\n\n C_{XY}[t,t+\\tau] = R_{XY}[t,t+\\tau] - E[X(t)]E[Y(t+\\tau)] <\/span>\n\n\n C_{XY}[t,t+\\tau] = R_{XY}[t,t+\\tau] - 0 <\/span>\n\n\n C_{XY}[t,t+\\tau] = \\frac{\\sin (\\omega \\tau)}{2} <\/span>\n\n\n\n
\n\n\n\n

X(t) y Y(t) son procesos con correlaci\u00f3n, pues su covarianza cruzada no es igual a cero para todas las selecciones de muestras de tiempo. Sin embargo, X(t1<\/sub>) y Y(t2<\/sub>) son variables aleatorias no correlacionadas para t1<\/sub> y t2<\/sub> dado que \u03c9( t2<\/sub>  - t1<\/sub> ) = k \u03c0, donde k es cualquier n\u00famero entero.<\/p>\n\n\n\n

\n\n\n\n

Los valores de mas medias de X(t) = Y(t) =0 son constantes<\/p>\n\n\n R_{X}[t,t+\\tau] = E[X(t) X(t+\\tau)] <\/span>\n\n\n =E[\\cos (\\omega t + \\theta) \\cos(\\omega (t+\\tau) + \\theta)] <\/span>\n\n\n =E\\Big[\\frac{1}{2} \\Big[ \\cos [(\\omega t + \\theta) -(\\omega (t+\\tau) + \\theta) ] + \\cos[(\\omega t + \\theta)+(\\omega (t+\\tau) + \\theta)] \\Big] \\Big] <\/span>\n\n\n =\\frac{1}{2}E\\Big[ \\cos (\\omega \\tau ) + \\cos(2\\omega t + \\omega \\tau + 2\\theta)] \\Big] <\/span>\n\n\n =\\frac{1}{2}E\\Big[ \\cos (\\omega \\tau )\\Big] +\\frac{1}{2}E\\Big[ \\cos(2\\omega t + \\omega \\tau + 2\\theta)] \\Big] <\/span>\n\n\n =\\frac{\\cos (\\omega \\tau )}{2} \\cos (\\omega \\tau ) +0 <\/span>\n\n\n\n

La autocorrelaci\u00f3n depende solo de las diferencias de tiempo \u03c4 = t2<\/sub>-t1<\/sub><\/p>\n\n\n\n

El proceso X(t) clasifica como Estacionario en el sentido amplio.<\/p>\n\n\n\n

Tarea<\/strong><\/em>: Revisar la autocorrelaci\u00f3n para Y(t) para verificar si clasifica como WSS.<\/p>\n","protected":false},"excerpt":{"rendered":"

Ejercicio: 2Eva2017TII_T2 Covarianza X, Y Tema 2. \u03b8 es una variable aleatoria uniforme, distribuida en el rango [-\u03c0 \u03c0]. valor esperados X(t) valor esperado Y(t) Correlaci\u00f3n X(t) y Y(t) El primer t\u00e9rmino no contiene la variable aleatorioa \u0398, por lo que se comporta como una constante para el valor esperado. X(t) y Y(t) son procesos […]<\/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":[203],"tags":[58,237],"class_list":["post-1697","post","type-post","status-publish","format-standard","hentry","category-stp-ejemplos","tag-ejemplos-python","tag-pestocasticos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1697","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=1697"}],"version-history":[{"count":4,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1697\/revisions"}],"predecessor-version":[{"id":23552,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1697\/revisions\/23552"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=1697"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=1697"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=1697"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}