Problem 1 Joint Distribution of Presidential Party Affiliation and Party Control of House of Representatives, 1860-1996 Democratic Control |Republican Control Total of House (Y = 0) of House (Y = 1) Democratic 0.412 0.030 President (X = 0) Republican 0.176 0.382 President (X = 1) Total (a) Interpret one of the joint probabilities and one of the marginal probabilities. (b) Compute E(X). How does this differ from E(X|Y = 0)? Explain. (c) If you picked one of the Republican presidents at random, what is the probability that during his term the Democrats had control of the House? (d) What would the joint distribution look like under independence? Check your results by calculating the two conditional distributuions and compare these to the marginal distribution. Problem 2 Let Y be a Bernoulli random variable with parameter p. (a) Find first four moments of Y (b) Find first four centered moments of Y (c) For which value of p is Var[Y] the largest? (d) Find skewness and kurtosis of Y. (e) For which value of p is skewness=1. Explain