Let Y = X + N, where X is the input, N is the noise, and Y is the output of a system. Assume that X and N are independent random variables. It is given that E[X] = 0, Var[X] = σ 2 X, E[N] = 0, Var[N] = σ 2 N . (a) Find the correlation coefficient rho between the input X and the output Y . (b) Suppose we estimate the input X by a linear function g(Y ) = aY . Find the value of a that minimizes the mean squared error E[(X − aY ) 2 ]. (c) Express the resulting mean squared error in terms of η = σ 2 X/σ2 N .