Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Beta-binomial distribution ▷ Cumulative distribution 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 cumulative distribution function of $X$ is

\[\label{eq:betabin-cdf} F_X(x) = \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \frac{\Gamma(n+1)}{\Gamma(\alpha+\beta+n)} \cdot \sum_{i=0}^{x} \frac{\Gamma(\alpha+i) \cdot \Gamma(\beta+n-i)}{\Gamma(i+1) \cdot \Gamma(n-i+1)}\]

where $\mathrm{B}(x,y)$ is the beta function and $\Gamma(x)$ is the gamma function.

Proof: The cumulative distribution function is defined as

\[\label{eq:cdf} F_X(x) = \mathrm{Pr}(X \leq x)\]

which, for a discrete random variable, evaluates to

\[\label{eq:cdf-disc} F_X(x) = \sum_{i=-\infty}^{x} f_X(i) \; .\]

With the probability mass function of the beta-binomial distribution, this becomes

\[\label{eq:betabin-cdf-s1} F_X(x) = \sum_{i=0}^{x} {n \choose i} \cdot \frac{\mathrm{B}(\alpha+i,\beta+n-i)}{\mathrm{B}(\alpha,\beta)} \; .\]

Using the expression of binomial coefficients in terms of factorials

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

the relationship between factorials and the gamma function

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

and the link between gamma function and beta function

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

equation \eqref{eq:betabin-cdf-s1} can be further developped as follows:

\[\label{eq:betabin-cdf-s2} \begin{split} F_X(x) &\overset{\eqref{eq:bincoeff-facts}}{=} \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \sum_{i=0}^{x} \frac{n!}{i! \, (n-i)!} \cdot \mathrm{B}(\alpha+i,\beta+n-i) \\ &\overset{\eqref{eq:betafct-gamfct}}{=} \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \sum_{i=0}^{x} \frac{n!}{i! \, (n-i)!} \cdot \frac{\Gamma(\alpha+i) \cdot \Gamma(\beta+n-i)}{\Gamma(\alpha+\beta+n)} \\ &= \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \frac{n!}{\Gamma(\alpha+\beta+n)} \cdot \sum_{i=0}^{x} \frac{\Gamma(\alpha+i) \cdot \Gamma(\beta+n-i)}{i! \, (n-i)!} \\ &\overset{\eqref{eq:facts-gamfct}}{=} \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \frac{\Gamma(n+1)}{\Gamma(\alpha+\beta+n)} \cdot \sum_{i=0}^{x} \frac{\Gamma(\alpha+i) \cdot \Gamma(\beta+n-i)}{\Gamma(i+1) \cdot \Gamma(n-i+1)} \; . \end{split}\]

This completes the proof.


Metadata: ID: P366 | shortcut: betabin-cdf | author: JoramSoch | date: 2022-10-22, 05:28.