If X ~ Gamma(a, ), find E[X.a*], specifying the values of s and a (with a always positive) for which it exists. Suppose that K is a random variable that only takes values in {0, 1, 2, 3, . . .}. Show that 8 E[K] =>P{K > k}. k=1 Suppose that we start with three red balls and one green ball in a bag. On each turn, a ball is drawn uniformly at random from the bag. If the ball is green, it is returned to the bag together with an extra green ball, and play continues. If the ball drawn is red, the game ends. Let K be the number of draws up to and including the ending (red) draw. Find a simple expression in terms of k for the probability that Kk. Hence (using (b)) or otherwise, calculate the expected value of K. Does K have a finite second moment (i.e., finite variance)?