Expectation of Negative Binomial Distribution

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Theorem

First Form

Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$.


Then the expectation of $X$ is given by:

$E \left({X}\right) = \dfrac {n p} q$


Second Form

Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.


Then the expectation of $X$ is given by:

$\expect X = \dfrac n p$