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Expected Values, Negative Binomials and Maclaurin Series

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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

.

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