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Beta Distribution {\mathrm{B}(\alpha,\beta)}\!| cdf =| mean =| median =| mode = for | variance =| skewness =| kurtosis =| entropy =| mgf =| char =| }} In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval 1: -
where and are parameters that must be greater than zero. When the "constant" is included explicitly, the density looks like this: -
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where and are, respectively, the gamma function and the beta function. The special case of the beta distribution when and is the standard uniform distribution. The expected value and variance of a beta random variable with parameters and are given by the formulae: -
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On the other hand, with the expected value and variance of a beta random variable given, the parameters and are calculated by the formulae: -
\alpha = \operatorname{E}(X) \left( \frac{\operatorname{E}(X)}{\operatorname{var}(X)} \left- \operatorname{E}(X) \right - 1 \right), -
where and . Cumulative distribution function The cumulative distribution function is -
where is the incomplete beta function and is the regularized incomplete beta function. Shapes The beta function can take on different shapes depending on the values of the two parameters: - is the uniform distribution
- is symmetric about 1/2 (red & purple plots)
- is U-shaped (red plot)
- is strictly increasing (green plot)
- is strictly decreasing (blue plot)
- is unimodal (purple & black plots)
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