Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Beta-binomial distribution ▷ Probability mass function in terms of gamma function

Theorem: Let $X$ be a random variable following a beta-binomial distribution:

\[\label{eq:betabin} X \sim \mathrm{BetBin}(n,\alpha,\beta) \; .\]

Then, the probability mass function of $X$ can be expressed as

\[\label{eq:betabin-pmfitogf} f_X(x) = \frac{\Gamma(n+1)}{\Gamma(x+1) \, \Gamma(n-x+1)} \cdot \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \, \Gamma(\beta)} \cdot \frac{\Gamma(\alpha+x) \, \Gamma(\beta+n-x)}{\Gamma(\alpha+\beta+n)}\]

where $\Gamma(x)$ is the gamma function.

Proof: The probability mass function of the beta-binomial distribution is given by

\[\label{eq:betabin-pmf} f_X(x) = {n \choose x} \cdot \frac{\mathrm{B}(\alpha+x,\beta+n-x)}{\mathrm{B}(\alpha,\beta)} \; .\]

Note that the binomial coefficient can be expressed in terms of factorials

\[\label{eq:bincoeff-facts} {n \choose x} = \frac{n!}{x! \, (n-x)!} \; ,\]

that factorials are related to the gamma function via $n! = \Gamma(n+1)$

\[\label{eq:facts-gamfct} \frac{n!}{x! \, (n-x)!} = \frac{\Gamma(n+1)}{\Gamma(x+1) \, \Gamma(n-x+1)}\]

and that the beta function is related to the gamma function via

\[\label{eq:betafct-gamfct} \mathrm{B}(\alpha,\beta) = \frac{\Gamma(\alpha) \, \Gamma(\beta)}{\Gamma(\alpha+\beta)} \; .\]

Applying \eqref{eq:bincoeff-facts}, \eqref{eq:facts-gamfct} and \eqref{eq:betafct-gamfct} to \eqref{eq:betabin-pmf}, we get

\[\label{eq:betabin-pmfitogf-qed} f_X(x) = \frac{\Gamma(n+1)}{\Gamma(x+1) \, \Gamma(n-x+1)} \cdot \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \, \Gamma(\beta)} \cdot \frac{\Gamma(\alpha+x) \, \Gamma(\beta+n-x)}{\Gamma(\alpha+\beta+n)} \; .\]

This completes the proof.


Metadata: ID: P365 | shortcut: betabin-pmfitogf | author: JoramSoch | date: 2022-10-20, 08:56.