1. Consider two coins: one with probability of Heads equal to 1/2 and another with probability of Heads equal to 1/3. Assume that you randomly' select one of the two coins and start flipping it until you get the first Head. Define the following RVs: U is the probability of Heads of the selected coin, X is the indicator variable that the first flip of the selected coin is Heads, Y is the number of flips of the selected coin until you get the first Head. Note that the conditional distribution of X|U is Bernoulli(U), and the conditional dis- tribution of Y|U is Geometric (U). (a) Find the unconditional mean of X using the Law of Total Expectation (Theorem. 3.5.2 in E&R). (Hint: You do not have to derive the conditional expectation E(X|U). Instead, you can get it directly from the formula for the mean of the Bernoulli distribution.) (b) Find the unconditional variance of X using the Law of Total Variance (Theorem. 3.5.6 in E&R). (Hint: You do not have to derive the conditional variance V(X|U). Instead, you can get it directly from the formula for the variance of the Bernoulli distribution.) (c) Find the unconditional mean of Y using the Law of Total Expectation. (Hint: You do not have to derive the conditional expectation E(Y]U). Instead, you can get it directly from the formula for the mean of the Geometric distribution.) (d) Find the unconditional variance of Y using the Law of Total Variance. (Hint: You do not have to derive the conditional variance V(Y|U). Instead, you can get it directly from the formula for the variance of the Geometric distribution.)