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where α and β are parameters that must be greater than zero.
When the "constant" is included explicitly, the density looks like this:
The special case of the beta distribution when α = 1 and β = 1 is the standard uniform distribution.
The kurtosis excess is:
where and .
Cumulative distribution function
The beta function can take on different shapes depending on the values of the two parameters:
- α = β = 1 is the uniform distribution
- α = β 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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