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

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

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

Then, the cumulative distribution function of $X$ is

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

where $\mathrm{B}(a,b)$ is the beta function and $\mathrm{B}(x;a,b)$ is the incomplete gamma function.

Proof: The probability density function of the beta distribution is:

\[\label{eq:beta-pdf} f_X(x) = \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1} \; .\]

Thus, the cumulative distribution function is:

\[\label{eq:beta-cdf-app} \begin{split} F_X(x) &= \int_{0}^{x} \mathrm{Bet}(z; \alpha, \beta) \, \mathrm{d}z \\ &= \int_{0}^{x} \frac{1}{\mathrm{B}(\alpha, \beta)} \, z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \\ &= \frac{1}{\mathrm{B}(\alpha, \beta)} \int_{0}^{x} z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \; . \end{split}\]

With the definition of the incomplete beta function

\[\label{eq:inc-beta-fct} \mathrm{B}(x;a,b) = \int_{0}^{x} t^{a-1} \, (1-t)^{b-1} \, \mathrm{d}t \; ,\]

we arrive at the final result given by equation \eqref{eq:beta-cdf}:

\[\label{eq:beta-cdf-qed} F_X(x) = \frac{\mathrm{B}(x; \alpha, \beta)}{\mathrm{B}(\alpha, \beta)} \; .\]
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Metadata: ID: P195 | shortcut: beta-cdf | author: JoramSoch | date: 2020-11-19, 08:01.