4. Conditional Expectation as Minimum Mean Squared Error Estimator

(from lecture notes of STAT 901 at the University of Waterloo, by Prof. Don McLeish)

Let (X, Y ) ∼ p(x, y). Suppose E[X2 + Y 2 ] < ∞.

(a) What constant is consider to be the best fit to a random variable in the sense of smallest mean squared error? In other words, what is the value of c solving