Let x and z be zero-mean, jointly gaussian random variables with variances v2 and v2 , respectively, and correlation coefficient t. (a) express z as a linear combination of x and gaussian r. v. y that is independent of x. find the linear combination and the variance v2y of y in order that z has variance v2z , and x and z have correlation coefficient t. (b) use the result of (a) to you prove that e[x2 z2] = e[x2]e[z2] + 2e[xz]2 . (hint: you may find it to review the solution to problem 6b of problem set 9x.) (c) let x(t) be a zero-mean, wide-sense stationary gaussian random process with autocorrelation function rx (x). the output of a "square-law detector" is y(t) = x(t)2 . use the result of (b) to show that ry (x) = rx (0)2 + 2r2x (x).