naginterfaces.library.stat.prob_​hypergeom

naginterfaces.library.stat.prob_hypergeom(n, l, m, k)

prob_hypergeom returns the lower tail, upper tail and point probabilities associated with a hypergeometric distribution.

For full information please refer to the NAG Library document for g01bl

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01blf.html

Parameters

nint

The parameter $n$ of the hypergeometric distribution.

lint

The parameter $l$ of the hypergeometric distribution.

mint

The parameter $m$ of the hypergeometric 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, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

On entry, $l=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $l\le n$.

(errno $2$)

On entry, $l=⟨value⟩$.

Constraint: $l\ge 0$.

(errno $3$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m\le n$.

(errno $3$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 0$.

(errno $4$)

On entry, $k=⟨value⟩$, $l=⟨value⟩$, $m=⟨value⟩$ and $l+m-n=⟨value⟩$.

Constraint: $k\ge l+m-n$.

(errno $4$)

On entry, $k=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $k\le m$.

(errno $4$)

On entry, $k=⟨value⟩$ and $l=⟨value⟩$.

Constraint: $k\le l$.

(errno $4$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $5$)

On entry, $n$ is too large to be represented exactly as a double precision number.

(errno $6$)

On entry, the variance $=\frac{lm(n-l)(n-m)}{n^2(n-1)}$ exceeds ${10}^6$.

Notes

Let $X$ denote a random variable having a hypergeometric distribution with parameters $n$, $l$ and $m$ ($n\ge l\ge 0$, $n\ge m\ge 0$). Then

$$\begin{array}{c}Prob\{X=k\}=\frac{\left(\begin{array}{c}m\\ k\end{array}\right)\left(\begin{array}{c}n-m\\ l-k\end{array}\right)}{\left(\begin{array}{c}n\\ l\end{array}\right)}\text{,}\end{array}$$

where $max(0,l-(n-m))\le k\le min(l,m)$, $0\le l\le n$ and $0\le m\le n$.

The hypergeometric distribution may arise if in a population of size $n$ a number $m$ are marked. From this population a sample of size $l$ is drawn and of these $k$ are observed to be marked.

The mean of the distribution $=\frac{lm}{n}$, and the variance $=\frac{lm(n-l)(n-m)}{n^2(n-1)}$.

prob_hypergeom computes for given $n$, $l$, $m$ 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 similar to the method for the Poisson distribution described in Knüsel (1986).

References

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