You go to the basketball courts in West 4 street to shoot hoops. Assume you make a successful shot with probability p [0, 1]. You shoot several times with independent outcomes.

(a) You decide to stop as soon as you make one successful shot. The total number of shots is a random variable X. Find the PMF of X.

(b) Now you decide to stop as soon as you make k successful shots. The total number of shots is a random variable Y . Find the PMF of Y .

(c) You realize that you have a 'hot hand'. That is P(shot n + 1 successful |shot n successful) = q and P(shot n + 1 successful |shot n not successful) = p, where n = 1, 2, and q > p. We assume P(shot 1 successful) = p.

i. Find the probability that you make 3 successful shots out of 4 total shots.

ii. You decide to stop as soon as you make 2 successful shots. The total number of shots is a random variable Z. Find the PMF of Z.