Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Beta distribution ▷ Median

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

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

Then, the median of $X$ is

\[\label{eq:beta-med} \mathrm{median}(X) = \mathrm{B}_{\alpha,\beta}^{-1}\left( \frac{\mathrm{B}(\alpha,\beta)}{2} \right) = \mathrm{I}_{1/2}^{-1}(\alpha,\beta)\]

where $\mathrm{B}(a,b)$ is the beta function, $\mathrm{B}_{a,b}^{-1}(y)$ is the inverse function of the incomplete beta function $\mathrm{B}(x; a, b)$ and $\mathrm{I}_y^{-1}(a,b)$ is the inverse function of the regularized incomplete beta function.

Proof: The median is the value at which the cumulative distribution function is $1/2$:

\[\label{eq:median} F_X(\mathrm{median}(X)) = \frac{1}{2} \; .\]

The cumulative distribution function of the beta distribution is

\[\label{eq:beta-cdf} F_X(x) = \frac{\mathrm{B}(x; \alpha, \beta)}{\mathrm{B}(\alpha, \beta)}\]

where the incomplete beta function $\mathrm{B}(x; a, b)$ is given by

\[\label{eq:inc-beta-fct} \mathrm{B}(x; a, b) = \int_0^x t^{a-1} (1-t)^{b-1} \, \mathrm{d}t \; .\]

Thus, the inverse CDF, i.e. the quantile function, is

\[\label{eq:beta-cdf-inv} x = \mathrm{B}_{\alpha,\beta}^{-1}(p \, \mathrm{B}(\alpha,\beta))\]

where $\mathrm{B}_{a,b}^{-1}(y)$ is the inverse function of $\mathrm{B}(x; a, b)$. Setting $p = 1/2$, we obtain:

\[\label{eq:beta-med-qed} \mathrm{median}(X) = \mathrm{B}_{\alpha,\beta}^{-1}\left( \frac{\mathrm{B}(\alpha,\beta)}{2} \right) \; .\]

Alternatively, the cumulative distribution function may be written as

\[\label{eq:beta-cdf-alt} F_X(x) = \mathrm{I}_x(a,b)\]

using the regularized incomplete beta function

\[\label{eq:reg-inc-beta-fct} \mathrm{I}_x(a,b) = \frac{\mathrm{B}(x; \alpha, \beta)}{\mathrm{B}(\alpha, \beta)}\]

in which case the inverse CDF is

\[\label{eq:beta-cdf-inv-alt} x = \mathrm{I}_p^{-1}(a,b) \; ,\]

such that the median becomes

\[\label{eq:beta-med-qed-alt} \mathrm{median}(X) = \mathrm{I}_{1/2}^{-1}(\alpha,\beta) \; .\]
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Metadata: ID: P519 | shortcut: beta-med | author: JoramSoch | date: 2025-10-24, 13:45.