Question 1 (10 points) Let X be a Bernoulli random variable with P[X = 1] =p (0,1), and let Z = V2+ X. (a) (5 points) Calculate E[Z]. (b) (5 points) Calculate V[Z]. Question 2 (20 points) Let Y, Y2 and Y3 be independent and identically distributed random variables with expectation E[Y] =u and variance V[Y;] = o for i = 1, 2, 3. Suppose we want to estimate the expectation H, and we propose to combine our three observations Y, Y2 and Y3 by defining a weighted average: Yu = wiY1 +w2Y2 + w3Y3 where wi, w2 and w3 are constants between 0 and 1. (a) (7 points) Calculate E[W]. What condition do the weights wi, w2 and w3 need to satisfy for Yw to be an unbiased estimator for u? (b) (6 points) Calculate V[w]. Using the condition you found on (a), find the set of weights that minimize the variance of Yw. Justify your answer. Now suppose that observations Y, Y, and Yz are independent and have the same mean u, but they have different variances. Specifically, V[Y1] = V[Y3] 1.502 and V[Y2] = o, so that the variances of Y and Y3 are 1.5 times as large as the variance of Y2. Act Got 1