naginterfaces.library.stat.prob_​binomial

naginterfaces.library.stat.prob_binomial(n, p, k)

prob_binomial returns the lower tail, upper tail and point probabilities associated with a binomial distribution.

For full information please refer to the NAG Library document for g01bj

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01bjf.html

Parameters

nint

The parameter $n$ of the binomial distribution.

pfloat

The parameter $p$ of the binomial 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, $p=⟨value⟩$.

Constraint: $p<1.0$.

(errno $2$)

On entry, $p=⟨value⟩$.

Constraint: $p>0.0$.

(errno $3$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $3$)

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

Constraint: $k\le n$.

(errno $4$)

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

(errno $5$)

On entry, the variance $(=np(1-p))$ exceeds ${10}^6$.

Notes

Let $X$ denote a random variable having a binomial distribution with parameters $n$ and $p$ ($n\ge 0$ and $0<p<1$). Then

$$\begin{array}{c}Prob\{X=k\}=\left(\begin{array}{c}n\\ k\end{array}\right)p^k{(1-p)}^{n-k}\text{,}k=0,1,\dots ,n\text{.}\end{array}$$

The mean of the distribution is $np$ and the variance is $np(1-p)$.

prob_binomial computes for given $n$, $p$ 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