Suppose that there are two random variables X and Y. Suppose we know the joint distribution of Y and X. We would like to use X to predict Y. Our prediction is therefore a function of X , denote as m(X). If we restrict m(X) to have the linear form, Note that there is no intercept in m(X). Now we ask the question \"What is the optimal prediction function we can get?\" i.e. to nd the optimal value of 61 (denoted by 51') in m(X) = lX that minimizes the mean squared error 51' = arg1%inEX,y[(Y 51902] Prove that the optimal solution is 5* _ Cov(X Y) + 1E(X )]E Y) _]E(XY) 1 _ Var(X )+ (11300)2 _ 1E(X2) ( Note that if lE(X) = lE(Y) 0 then 61 Cov X,Y)/Var(X)