# NAG Library Function Document

## 1Purpose

nag_poisson_dist (g01bkc) returns the lower tail, upper tail and point probabilities associated with a Poisson distribution.

## 2Specification

 #include #include
 void nag_poisson_dist (double rlamda, Integer k, double *plek, double *pgtk, double *peqk, NagError *fail)

## 3Description

Let $X$ denote a random variable having a Poisson distribution with parameter $\lambda$ $\left(>0\right)$. Then
 $ProbX=k=e-λλkk! , k=0,1,2,…$
The mean and variance of the distribution are both equal to $\lambda$.
nag_poisson_dist (g01bkc) computes for given $\lambda$ and $k$ the probabilities:
 $plek=ProbX≤k pgtk=ProbX>k peqk=ProbX=k .$
The method is described in Knüsel (1986).

## 4References

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

## 5Arguments

1:    $\mathbf{rlamda}$doubleInput
On entry: the parameter $\lambda$ of the Poisson distribution.
Constraint: $0.0<{\mathbf{rlamda}}\le {10}^{6}$.
2:    $\mathbf{k}$IntegerInput
On entry: the integer $k$ which defines the required probabilities.
Constraint: ${\mathbf{k}}\ge 0$.
3:    $\mathbf{plek}$double *Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
4:    $\mathbf{pgtk}$double *Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
5:    $\mathbf{peqk}$double *Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT
On entry, ${\mathbf{rlamda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rlamda}}\le {10}^{6}$.
NE_REAL_ARG_LE
On entry, ${\mathbf{rlamda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rlamda}}>0.0$.

## 7Accuracy

Results are correct to a relative accuracy of at least ${10}^{-6}$ on machines with a precision of $9$ or more decimal digits, and to a relative accuracy of at least ${10}^{-3}$ on machines of lower precision (provided that the results do not underflow to zero).

## 8Parallelism and Performance

nag_poisson_dist (g01bkc) is not threaded in any implementation.

The time taken by nag_poisson_dist (g01bkc) depends on $\lambda$ and $k$. For given $\lambda$, the time is greatest when $k\approx \lambda$, and is then approximately proportional to $\sqrt{\lambda }$.

## 10Example

This example reads values of $\lambda$ and $k$ from a data file until end-of-file is reached, and prints the corresponding probabilities.

### 10.1Program Text

Program Text (g01bkce.c)

### 10.2Program Data

Program Data (g01bkce.d)

### 10.3Program Results

Program Results (g01bkce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017