Y ( t ) = a cos ( 2 π t + π 2 X ( t ) ) Y(t) = a \cos \big( 2\pi t + \frac{\pi}{2}X(t) \big) Y ( t ) = a cos ( 2 π t + 2 π X ( t ) ) c) Encuentre la media y autocorrelacion de Y(t)
E [ Y ( t 1 ) Y ( t 2 ) ] = E [ a cos ( 2 π t 1 + π 2 X ) . a cos ( 2 π t 2 + π 2 X ) ] 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] E [ Y ( t 1 ) Y ( t 2 ) ] = E [ a cos ( 2 π t 1 + 2 π X ) . a cos ( 2 π t 2 + 2 π X ) ] g ( x ) = a cos ( 2 π t 1 + π 2 X ) . a cos ( 2 π t 2 + π 2 X ) 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) g ( x ) = a cos ( 2 π t 1 + 2 π X ) . a cos ( 2 π t 2 + 2 π X ) Referencia: Valor esperado de funciones de variable aleatoria (León-García 3.3.1 p. 107E [ g ( x ) ] = ∑ k g ( x k ) p x ( X k ) E[g(x)] = \sum_k g(x_k)p_x(X_k) E [ g ( x ) ] = k ∑ g ( x k ) p x ( X k ) = [ a cos ( 2 π t 1 + π 2 ( − 1 ) ) . a cos ( 2 π t 2 + π 2 ( − 1 ) ) ] 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} + = [ a cos ( 2 π t 1 + 2 π ( − 1 ) ) . a cos ( 2 π t 2 + 2 π ( − 1 ) ) ] 2 1 + + [ a cos ( 2 π t 1 + π 2 ( 1 ) ) . a cos ( 2 π t 2 + π 2 ( 1 ) ) ] 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} + [ a cos ( 2 π t 1 + 2 π ( 1 ) ) . a cos ( 2 π t 2 + 2 π ( 1 ) ) ] 2 1 = a 2 2 cos ( 2 π t 1 − π 2 ) cos ( 2 π t 2 − π 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) + = 2 a 2 cos ( 2 π t 1 − 2 π ) cos ( 2 π t 2 − 2 π ) + + a 2 2 cos ( 2 π t 1 + π 2 ) cos ( 2 π t 2 + π 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) + 2 a 2 cos ( 2 π t 1 + 2 π ) cos ( 2 π t 2 + 2 π ) = a 2 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) + = \frac{a^2}{2} \sin (2\pi t_1) \sin (2\pi t_2) + = 2 a 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) + + a 2 2 [ − sin ( 2 π t 1 ) ] [ − sin ( 2 π t 2 ) ] + \frac{a^2}{2}[-\sin(2\pi t_1)][-\sin (2\pi t_2)] + 2 a 2 [ − sin ( 2 π t 1 ) ] [ − sin ( 2 π t 2 ) ] = a 2 2 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) = \frac{a^2}{2} 2 \sin (2\pi t_1) \sin( 2\pi t_2) = 2 a 2 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) = a 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) = a^2 \sin ( 2\pi t_1) \sin(2\pi t_2) = a 2 sin ( 2 π t 1 ) sin ( 2 π t 2 ) = a 2 2 [ cos ( 2 π t 1 − 2 π t 2 ) − cos ( 2 π t 1 + 2 π 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] = 2 a 2 [ cos ( 2 π t 1 − 2 π t 2 ) − cos ( 2 π t 1 + 2 π t 2 ) ] E [ Y ( t 1 ) Y ( t 2 ) ] = a 2 2 [ cos ( 2 π t 1 − 2 π t 2 ) − cos ( 2 π t 1 + 2 π t 2 ) ] 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] E [ Y ( t 1 ) Y ( t 2 ) ] = 2 a 2 [ cos ( 2 π t 1 − 2 π t 2 ) − cos ( 2 π t 1 + 2 π t 2 ) ]
