Expectation of Negative Binomial Distribution/Second Form

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Theorem

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:

$\expect X = \dfrac n p$


Proof

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

$\map {\Pi_X} s = \paren {\dfrac {p s} {1 - q s} }^n$

where $q = 1 - p$.


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

$\expect X = \map {\Pi'_X} 1$


We have:

\(\ds \map {\Pi'_X} s\) \(=\) \(\ds \map {\frac \d {\d s} } {\frac {p s} {1 - q s} }^n\)
\(\ds \) \(=\) \(\ds n p \paren {\frac {\paren {p s}^{n - 1} } {\paren {1 - q s}^{n + 1} } }\) First Derivative of PGF of Negative Binomial Distribution/Second Form


Plugging in $s = 1$:

\(\ds \map {\Pi'_X} 1\) \(=\) \(\ds n p \paren {\frac {p^{n - 1} } {\paren {1 - q}^{n + 1} } }\)
\(\ds \) \(=\) \(\ds n p \paren {\frac {p^{n - 1} } {p^{n + 1} } }\) as $p = 1 - q$
\(\ds \leadsto \ \ \) \(\ds \expect X\) \(=\) \(\ds \frac n p\) simplifying

$\blacksquare$


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