esway Problem 1 In class, we showed that E[aX] = aE[X] and E[X + Y] = E[X] + E[Y] for any random variables X and Y and a constant a. If X and Y are independent random variables then E[XY] = E[X]E[Y]. Using these three properties show that: (a) If X and Y are independent then Cov(X, Y) = 0 (b) Var[X] = E[X?] - (E[X])2 (c) Var[X + Y] = Var[X] + Var[Y] + 2Cov(X, Y) (d) If X and Y are independent then Var[X + Y] = Var[X] + Var[Y] (e) Cov(X, Y) = E[XY] - E[X]ELY] (f) Cov(a + bX + cY, W) = bCov(X, W) + cCov(Y, W) for random variables X, Y and W (g) If E[Y X] = My then Cov(X, Y) = Corr(X, Y) = 0. Problem 2 Assume three random variables X, Y and U satisfy the following two conditions: Y = Bo+ BIX +U, and E[U|X] = 0. If you know E[X], E[Y], Var [X], Cov(X, Y) (a) find conditions when there exists unique values Bo and B, and find it (b) if you also know Var[Y], find Var[U] (c) repeat (a) and (b) for Y = Bo + BIX1 + 2X2 + U and E[U|X1, X2] = 0 if you know E[Y], E[X]], E[X2], Cov(Y, X]), Cov(Y, X2), Var[X]], Var[X2], Cov(X1, X2) and Var[Y]. Problem 3 Calculate the following probabilities using the standard normal distribution. Sketch the probability distribution in each case, shading in the area of the calculated probability. (a) P(Z 1.96). (d) P(Z