Val wants to apply random variables to analyze his basketball skills. After shooting two basketballs, let X be the independent Bernoulli random variable that Val makes the FIRST shot with parameter 1/2 and Y be the independent Bernoulli random variable that Val misses the SECOND shot with parameter 1/2. Let a third random variable, Z, equal the remainder of (X+Y)/2.

(a) Prove that Z will also be a Bernoulli random variable with parameter 1/2.

(b) Prove that the random variables X, Y, Z are pairwise independent but not mutually independent.

(c) By computing [X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that [X+Y+Z] = [X] + [Y] + [Z]