Let X equal the number of flips of a fair coin that are required to observe the

Question:

Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips.
(a) Find the pmf of X.
(b) Find the moment-generating function of X.
(c) Use the mgf to find the values of
(i) The mean and
(ii) The variance of X.
(d) Find the values of
(i) P(X ≤ 3),
(ii) P(X ≥ 5), and
(iii) P(X = 3).