Question

Suppose a fair coin is tossed repeatedly and independently until the pair HT occurs (i.e., a head followed by a tail). Let X be the number of trials required to obtain the ordered pair HT.

• (a) Show that the probability mass function of X is given by p(x) = (x − 1)/2^x. (Hint: Draw a tree and note that at stage x, for x ≥ 2, there are precisely x − 1 paths in which an H was obtained at stage x − 1.)
• (b) Show that the moment-generating function of X is m(t) = e^2t/(e^t − 2)2.

Answer 1