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In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval [0, 1]:
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where a and b are parameters that must be greater than zero.
When the "constant" is included explicitly, the density looks like this:
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where Γ and B are respectively the gamma function and the
beta function.
The special case of the beta distribution, when a = 1 and b = 1, is the standard uniform distribution.
The expected value and variance of a beta random variable X with
parameters a and b are given by the formulae:
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On the other hand, with the expected value and variance of a beta random
variable X given, the parameters a and b are calculated by the formulae:
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where 0 < E(X) < 1 and 0 < var(X) < E(X) (1 − E(X)).
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