Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Beta-binomial distribution ▷ Probability mass 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 $X$ is

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

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

Proof: A beta-binomial random variable is defined as a binomial variate for which the success probability is following a beta distribution:

\[\label{eq:betabin-bin-beta} \begin{split} X \mid p &\sim \mathrm{Bin}(n, p) \\ p &\sim \mathrm{Bet}(\alpha, \beta) \; . \end{split}\]

Thus, we can combine the law of marginal probability and the law of conditional probability to derive the probability of $X$ as

\[\label{eq:betabin-pmf-s1} \begin{split} p(x) &= \int_\mathcal{P} \mathrm{p}(x,p) \, \mathrm{d}p \\ &= \int_\mathcal{P} \mathrm{p}(x \vert p) \, \mathrm{p}(p) \, \mathrm{d}p \; . \end{split}\]

Now, we can plug in the probability mass function of the binomial distribution and the probability density function of the beta distribution to get

\[\label{eq:betabin-pmf-s2} \begin{split} p(x) &= \int_0^1 {n \choose x} \, p^x \, (1-p)^{n-x} \cdot \frac{1}{\mathrm{B}(\alpha, \beta)} \, p^{\alpha-1} \, (1-p)^{\beta-1} \, \mathrm{d}p \\ &= {n \choose x} \cdot \frac{1}{\mathrm{B}(\alpha, \beta)} \, \int_0^1 p^{\alpha+x-1} \, (1-p)^{\beta+n-x-1} \, \mathrm{d}p \\ &= {n \choose x} \cdot \frac{\mathrm{B}(\alpha+x,\beta+n-x)}{\mathrm{B}(\alpha, \beta)} \, \int_0^1 \frac{1}{\mathrm{B}(\alpha+x,\beta+n-x)} \, p^{\alpha+x-1} \, (1-p)^{\beta+n-x-1} \, \mathrm{d}p \; . \end{split}\]

Finally, we recognize that the integrand is equal to the probability density function of a beta distribution and because probability density integrates to one, we have

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

This completes the proof.


Metadata: ID: P364 | shortcut: betabin-pmf | author: JoramSoch | date: 2022-10-20, 08:56.