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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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.