Beta prime distribution

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Beta Prime
Probability density function
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Cumulative distribution function
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Parameters shape (real)
shape (real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)

where is the Gauss's hypergeometric function 2F1

Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

where is a Beta function. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom.

The mode of a variate distributed as is . Its mean is if (if the mean is infinite, in other words it has no well defined mean) and its variance is if .

If X is a variate then is a variate.

If X is a then and are and variates.

If X and Y are and variates, then is a variate.

References

MathWorld article

eo:Vikipedio:Projekto matematiko/Β prima distribuo fa:توزیع بتا پریم


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