Expectation of Negative Binomial Distribution/First Form

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Theorem

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:

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


Proof

From Probability Generating Function of Negative Binomial Distribution (First Form), we have:

$\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$

where $q = 1 - p$.


From Expectation of Discrete Random Variable from PGF, we have:

$E \left({X}\right) = \Pi'_X \left({1}\right)$


We have:

\(\ds \Pi'_X \left({s}\right)\) \(=\) \(\ds \frac {\mathrm d} {\mathrm ds} \left({\dfrac q {1 - ps} }\right)^n\)
\(\ds \) \(=\) \(\ds \dfrac {n p} q \left({\dfrac q {1 - ps} }\right)^{n+1}\) First Derivative of PGF of Negative Binomial Distribution/First Form


Plugging in $s = 1$:

\(\ds \Pi'_X \left({1}\right)\) \(=\) \(\ds \frac {n p} q \left({\frac q {1 - p} }\right)^{n+1}\)
\(\ds \) \(=\) \(\ds \frac {n p} q \left({\frac q q }\right)^{n+1}\) as $p = 1 - q$
\(\ds \implies \ \ \) \(\ds E \left({X}\right)\) \(=\) \(\ds \frac {n p} q\) simplifying

$\blacksquare$