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Evaluates the PDF of the generalized Beta distribution.

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Interface

#include <codecogs/statistics/distributions/continuous/beta/pdf.h>

using namespace Statistics::Distributions::Continuous::Beta;

double PDF(double x, double a, double b)

The density of the Beta distribution is defined by

$P(x) = \frac{(1-x)^{\beta-1} x^{\alpha-1}}{B(\alpha, \beta)}$

(1)

$P(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} (1-x)^{\beta-1} x^{\alpha-1}$

(2)

When $\alpha$ and $\beta$ are equal the function is symmetric about x=0.5, with it becoming increasingly skewed with larger and larger differences. $\alpha = \beta =1$ is a uniform distribution, while $\alpha = 1, \beta =2$ produces a straight line of gradient -2, and $\alpha = 2, \beta =1$ produces a straight line with a gradient of 2.

As a general rule values of $\alpha >1, \beta >1$ produces a closed distributions, i.e \graph x=0:0.99 a=3 b=3:9:4 .size=medium

While values of $\alpha <1, \beta <1$ produce cupped shaped distributions which tend to infinity at either extreme, i.e. \graph x=0.1:0.9 a=0.2:0.8:7 b=0.5 .size=medium

References

Cephes Math Library Release 2.8: June, 2000

Parameters

quantile in the range 0<=x<=1.

1st degrees of freedom. a>=0.

2nd degrees of freedom. b>=0..

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Result –