naginterfaces.library.stat.prob_​binomial_​vector

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

prob_binomial_vector returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

For full information please refer to the NAG Library document for g01sj

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01sjf.html

Parameters

nint, array-like, shape $\left(\text{ln}\right)$

$n_i$, the first parameter of the binomial distribution.

pfloat, array-like, shape $\left(\text{lp}\right)$

$p_i$, the second parameter of the binomial distribution.

kint, array-like, shape $\left(\text{lk}\right)$

$k_i$, the integer which defines the required probabilities.

Returns

plekfloat, ndarray, shape $(max(\text{ln},\text{lp}))$

$Prob\{X_i\le k_i\}$, the lower tail probabilities.

pgtkfloat, ndarray, shape $(max(\text{ln},\text{lp}))$

$Prob\{X_i>k_i\}$, the upper tail probabilities.

peqkfloat, ndarray, shape $(max(\text{ln},\text{lp}))$

$Prob\{X_i=k_i\}$, the point probabilities.

ivalidint, ndarray, shape $(max(\text{ln},\text{lp}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, $n_i<0$.

$ivalid[i-1]=2$

On entry, $p_i\le 0.0$, or, $p_i\ge 1.0$.

$ivalid[i-1]=3$

On entry, $k_i<0$, or, $k_i>n_i$.

$ivalid[i-1]=4$

On entry, $n_i$ is too large to be represented exactly as a real number.

$ivalid[i-1]=5$

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

Raises

NagValueError

(errno $2$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ln}>0$.

(errno $3$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lp}>0$.

(errno $4$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lk}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $n$, $p$ or $k$ was invalid.

Check $ivalid$ for more information.

Notes

Let $X=\{X_i:i=1,2,\dots ,m\}$ denote a vector of random variables each having a binomial distribution with parameters $n_i$ and $p_i$ ($n_i\ge 0$ and $0<p_i<1$). Then

$$\begin{array}{c}Prob\{X_i=k_i\}=\left(\begin{array}{c}{n}_i\\ k_i\end{array}\right)p_i^{k_i}{(1-p_i)}^{n_i-k_i}\text{,}{k}_i=0,1,\dots ,n_i\text{.}\end{array}$$

The mean of the each distribution is given by $n_i{p}_i$ and the variance by $n_i{p}_i(1-p_i)$.

prob_binomial_vector computes, for given $n_i$, $p_i$ and $k_i$, the probabilities: $Prob\{X_i\le k_i\}$, $Prob\{X_i>k_i\}$ and $Prob\{X_i=k_i\}$ using an algorithm similar to that described in Knüsel (1986) for the Poisson distribution.

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

References

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