The beta distribution is a continuous probability distribution related to the gamma distribution with a probability density function defined in the interval [0, 1]. Beta distributions have two free parameters, denoted here as and .

 

Beta Function

 

 

where is the complete gamma function.

 

Incomplete Beta Function

 

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Note that the restriction above is used to distinguish this function from the beta function.

 

Normalized Incomplete Beta Function

 

 

Probability Density Function

 

 

Cumulative Distribution Function