Definition:Beta Distribution

Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \closedint 0 1$.

$X$ is said to have a beta distribution if and only if it has probability density function:

$\map {f_X} X = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$

for $\alpha, \beta > 0$, where $\Beta$ denotes the beta function.

This is written:

$X \sim \BetaDist \alpha \beta$

Also see

• Definition:U-Shaped Distribution
• Definition:Uniform Distribution

• Results about the beta distribution can be found here.

Technical Note

The $\LaTeX$ code for \(\BetaDist {\alpha} {\beta}\) is \BetaDist {\alpha} {\beta} .

When the arguments are single characters, it is usual to omit the braces:

\BetaDist \alpha \beta

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): beta distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): beta distribution

• Weisstein, Eric W. "Beta Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BetaDistribution.html