Minimizing mean-squared error for random sequences

[rsingh628]

Hi all, I’m stuck on where to begin. For the question below, my approach is simply to substitute the $S_k (hat)$ into the MSE formula, get an expression in terms of $c_0$ and take the derivative and setting it to 0 to find the min $c_0$. I guess that is where I’m stuck in that I have the MSE in terms of expectations. If I substitute $S_k (hat)$ into the formula and distribute terms, I get $MSE = (c_0)^2E(X_k^2) - 2c_0E(X_kS_k) + E(S_k^2)$ so I’m struggling to find of those expectations. The $S_k$ are iid Gaussian with mean 1 and variance 1, but when I try to calculate, say, $E(X_k)$ , I get 0 since all $S_k$ are independent. That doesn’t help much, so I guess my other issue is finding $E(X_k^2)$, $E(X_kS_k)$, $E(S_k^2)$ etc.

Any help would be much appreciated, thank you.