naginterfaces.library.stat.prob_​poisson

naginterfaces.library.stat.prob_poisson(rlamda, k)

prob_poisson returns the lower tail, upper tail and point probabilities associated with a Poisson distribution.

For full information please refer to the NAG Library document for g01bk

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g01/g01bkf.html

Parameters

rlamdafloat

The parameter $\lambda$ of the Poisson distribution.

kint

The integer $k$ which defines the required probabilities.

Returns

plekfloat

The lower tail probability, $Prob\{X\le k\}$.

pgtkfloat

The upper tail probability, $Prob\{X>k\}$.

peqkfloat

The point probability, $Prob\{X=k\}$.

Raises

NagValueError

(errno $1$)

On entry, $rlamda=⟨value⟩$.

Constraint: $rlamda>0.0$.

(errno $2$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $3$)

On entry, $rlamda=⟨value⟩$.

Constraint: $rlamda\le {10}^6$.

Notes

Let $X$ denote a random variable having a Poisson distribution with parameter $\lambda$ $(>0)$. Then

$$Prob\{X=k\}=e^{-\lambda }\frac{{\lambda }^k}{k!}\text{,}k=0,1,2,\dots$$

The mean and variance of the distribution are both equal to $\lambda$.

prob_poisson computes for given $\lambda$ and $k$ the probabilities:

$$\begin{array}{c}\begin{array}{c}plek=Prob\{X\le k\}\\ pgtk=Prob\{X>k\}\\ peqk=Prob\{X=k\}\text{.}\end{array}\end{array}$$

The method is described in Knüsel (1986).

References

Knüsel, L, 1986, Computation of the chi-square and Poisson distribution, SIAM J. Sci. Statist. Comput. (7), 1022–1036