Consider the experiment of observing a sequence of Bernoulli trials until exactly r successes occur. Let the r.v. X denote the number of trials needed to observe the rth success. The r.v. X is known as the negative binomial r.v. with parameter p, where p is the probability of a success at each trial.
The pmf of the negative binomial distribution is given by:
Use the Maclaurin series expansions of the negative binomial and its derivatives and to find E(X). Keep in mind that
Expected Values, Negative Binomials and Maclaurin Series are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.