c) Encuentre la media y autocorrelacion de Y(t)
E[Y(t_1) Y(t_2)] = E\big[a \cos \big( 2\pi t_1 + \frac{\pi}{2}X \big) . a \cos \big( 2\pi t_2 + \frac{\pi}{2}X\big)\big] g(x) = a \cos \big( 2\pi t_1 + \frac{\pi}{2}X \big) . a \cos \big( 2\pi t_2 + \frac{\pi}{2}X\big)Referencia: Valor esperado de funciones de variable aleatoria (León-García 3.3.1 p. 107
Si z =g(x)
E[g(x)] = \sum_k g(x_k)p_x(X_k)
tomando la pmd mostrada en el cálculo del valor esperado, se tiene entonces que:
= \big[ a \cos \big( 2\pi t_1 + \frac{\pi}{2}(-1)\big) . a \cos \big( 2\pi t_2 + \frac{\pi}{2}(-1)\big)\big] \frac{1}{2} +
+ \big[ a \cos \big( 2\pi t_1 + \frac{\pi}{2}(1)\big) . a \cos \big( 2\pi t_2 + \frac{\pi}{2}(1)\big)\big] \frac{1}{2}
= \frac{a^2}{2} \cos \big( 2\pi t_1 - \frac{\pi}{2}\big) \cos \big( 2\pi t_2 - \frac{\pi}{2}\big) +
+ \frac{a^2}{2}\cos \big( 2\pi t_1 + \frac{\pi}{2}\big) \cos \big( 2\pi t_2 + \frac{\pi}{2}\big)
= \frac{a^2}{2} \sin (2\pi t_1) \sin (2\pi t_2) +
+ \frac{a^2}{2}[-\sin(2\pi t_1)][-\sin (2\pi t_2)]
= \frac{a^2}{2} 2 \sin (2\pi t_1) \sin( 2\pi t_2)
= a^2 \sin ( 2\pi t_1) \sin(2\pi t_2)
= \frac{a^2}{2}\big[ \cos(2\pi t_1 - 2\pi t_2) - \cos(2\pi t_1 + 2\pi t_2) \big]
E[Y(t_1)Y(t_2)] = \frac{a^2}{2}\big[ \cos(2\pi t_1 - 2\pi t_2) - \cos(2\pi t_1 + 2\pi t_2) \big]