naginterfaces.library.stat.prob_​beta

naginterfaces.library.stat.prob_beta(x, a, b)

prob_beta computes the upper and lower tail probabilities and the probability density function of the beta distribution with parameters $a$ and $b$.

For full information please refer to the NAG Library document for g01ee

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01eef.html

Parameters

xfloat

$\beta$, the value of the beta variate.

afloat

$a$, the first parameter of the required beta distribution.

bfloat

$b$, the second parameter of the required beta distribution.

Returns

pfloat

The lower tail probability, $P(B\le \beta :a,b)$.

qfloat

The upper tail probability, $P(B\ge \beta :a,b)$.

pdffloat

The probability density function, $f(B:a,b)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge 0.0$.

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le 1.0$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $a>0.0$.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $b>0.0$.

Notes

The probability density function of the beta distribution with parameters $a$ and $b$ is:

$$f(B:a,b)=\frac{\Gamma (a+b)}{\Gamma \left(a\right)\Gamma \left(b\right)}{B}^{a-1}{(1-B)}^{b-1}\text{,}0\le B\le 1\text{;}a,b>0\text{.}$$

The lower tail probability, $P(B\le \beta :a,b)$ is defined by

$$P(B\le \beta :a,b)=\frac{\Gamma (a+b)}{\Gamma \left(a\right)\Gamma \left(b\right)}{\int }_0^{\beta }{B}^{a-1}{(1-B)}^{b-1}dB=I_{\beta }(a,b)\text{,}0\le \beta \le 1\text{;}a,b>0\text{.}$$

The function $I_x(a,b)$, also known as the incomplete beta function is calculated using specfun.beta_incomplete.

References

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth