naginterfaces.library.stat.prob_​gamma

naginterfaces.library.stat.prob_gamma(g, a, b, tail='L')

prob_gamma returns the lower or upper tail probability of the gamma distribution, with parameters $\alpha$ and $\beta$.

For full information please refer to the NAG Library document for g01ef

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01eff.html

Parameters

gfloat

$g$, the value of the gamma variate.

afloat

The parameter $\alpha$ of the gamma distribution.

bfloat

The parameter $\beta$ of the gamma distribution.

tailstr, length 1, optional

Indicates whether an upper or lower tail probability is required.

$tail=\text{'L'}$

The lower tail probability is returned, that is $P(G\le g:\alpha ,\beta )$.

$tail=\text{'U'}$

The upper tail probability is returned, that is $P(G\ge g:\alpha ,\beta )$.

Returns

pfloat

The lower or upper tail probability of the gamma distribution, with parameters $\alpha$ and $\beta$.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$ or $\text{'U'}$.

(errno $2$)

On entry, $g=⟨value⟩$.

Constraint: $g\ge 0.0$.

(errno $3$)

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

Constraint: $a>0.0$ and $b>0.0$.

Warns

NagAlgorithmicWarning

(errno $4$)

The algorithm has failed to converge in $⟨value⟩$ iterations. The probability returned should be a reasonable approximation to the solution.

Notes

The lower tail probability for the gamma distribution with parameters $\alpha$ and $\beta$, $P(G\le g)$, is defined by:

$$P(G\le g\text{;}\alpha ,\beta )=\frac{1}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}{\int }_0^g{G}^{\alpha -1}{e}^{-G/\beta }dG\text{,}\alpha >0.0\text{,}\beta >0.0\text{.}$$

The mean of the distribution is $\alpha \beta$ and its variance is $\alpha {\beta }^2$. The transformation $Z=\frac{G}{\beta }$ is applied to yield the following incomplete gamma function in normalized form,

$$P(G\le g\text{;}\alpha ,\beta )=P(Z\le g/\beta :\alpha ,1.0)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{g/\beta }{Z}^{\alpha -1}{e}^{-Z}dZ\text{.}$$

This is then evaluated using specfun.gamma_incomplete.

References

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