Second Derivative of PGF of Negative Binomial Distribution/Second Form
Jump to navigation
Jump to search
Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.
Then the second derivative of the PGF of $X$ with respect to $s$ is:
- $\dfrac {\d^2} {\d s^2} \map {\Pi_X} s = \paren {\dfrac {p s} {1 - q s} }^{n + 2} \paren {\dfrac {n \paren {n - 1} + 2 n q s} {\paren {p s^2}^2} }$
Proof
The Probability Generating Function of Negative Binomial Distribution (Second Form) is:
- $\map {\Pi_X} s = \paren {\dfrac {p s} {1 - q s} }^n$
We have that for a given negative binomial distribution, $n, p$ and $q$ are constant.
From First Derivative of PGF of Negative Binomial Distribution/Second Form:
\(\ds \frac \d {\d s} \map {\Pi_X} s\) | \(=\) | \(\ds n p \paren {\dfrac {\paren {p s}^{n - 1} } {\paren {1 - q s}^{n + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac n {p s^2} \paren {\frac {p s} {1 - q s} }^{n + 1}\) |
Thus we have:
\(\ds \frac {\d^2} {\d s^2} \map {\Pi_X} s\) | \(=\) | \(\ds \map {\frac \d {\d s} } {\frac n {p s^2} \paren {\frac {p s} {1 - q s} }^{n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac n {p s^2} \map {\frac \d {\d s} } {\paren {\frac {p s} {1 - q s} }^{n + 1} } + \map {\frac \d {\d s} } {\frac n {p s^2} } \paren {\frac {p s} {1 - q s} }^{n + 1}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac n {p s^2} \paren {\frac {n + 1} {p s^2} \paren {\frac {p s} {1 - q s} }^{n + 2} } + \map {\frac \d {\d s} } {\frac n {p s^2} } \paren {\frac {p s} {1 - q s} }^{n + 1}\) | First Derivative of PGF of Negative Binomial Distribution/Second Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac n {p s^2} \paren {\frac {n + 1} {p s^2} \paren {\frac {p s} {1 - q s} }^{n + 2} } + \paren {\frac {- 2 n} {p s^3} } \paren {\frac {p s} {1 - q s} }^{n + 1}\) | Power Rule for Derivatives where $n = -2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} } {p^2 s^4} \paren {\frac {p s} {1 - q s} }^{n + 2} + \paren {\frac {- 2 n} {p s^3} } \paren {\frac {p s} {1 - q s} }^{n + 1}\) | dismayingly messy algebra | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 1} \paren {\frac {n \paren {n + 1} } {p s^3} \paren {\frac 1 {1 - q s} } + \paren {\frac {- 2 n} {p s^3} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 1} \paren {\frac {n \paren {n + 1} - 2 n \paren {1 - q s} } {p s^3 \paren {1 - q s} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 1} \paren {\frac {n^2 + n - 2 n + 2 n q s} {p s^3 \paren {1 - q s} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 1} \paren {\frac {n^2 - n + 2 n q s} {p s^3 \paren {1 - q s} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 1} \paren {\frac {n \paren {n - 1} + 2 n q s} {p s^3 \paren {1 - q s} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 2} \paren {\frac {n \paren {n - 1} + 2 n q s} {p^2 s^4} }\) | multiplying top and bottom by $p s$ and gathering terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {p s} {1 - q s} }^{n + 2} \paren {\frac {n \paren {n - 1} + 2 n q s} {\paren {p s^2}^2} }\) | final tidy up |
$\blacksquare$
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |