1. Let X and Y be two independent random variables with densities p() and p(y), respectively. Show the following two properties: Ep(2.9)[X +aY] = Ep(2)[X] + aEp()[Y] (1) Varp(wy)[X +aY] = Varp(x)[X] +aVarp() [Y] (2) for any scalar constant a E R. Hint: use the definition of expectation and variance, Epts)[X] = [_p()zdt (3) varp(x) [X] = Ep(x) [X2] E) [X] (4)