Chapter 2.3, Problem 17E

Let X equal the number of flips of a fair coin that are required to observe heads—tails on consecutive flips.

(a) Find the pmf of X. HINE Draw a tree diagram.

(b) Show that the mgf of X is $M\left(t\right)=\frac{e^{2t}}{{\left(e^t-2\right)}^2}$.

(c) Use the mgf to find the values of (I) the mean and (ii) the variance of X.

(d) Find the values of (I) $P\left(X\le 3\right)$, (ii) $P\left(X\ge 5\right)$, and (iii) $P\left(X=3\right)$.