naginterfaces.library.stat.prob_​poisson_​vector

naginterfaces.library.stat.prob_poisson_vector(l, k)

prob_poisson_vector returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.

For full information please refer to the NAG Library document for g01sk

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01skf.html

Parameters

lfloat, array-like, shape $\left(\text{ll}\right)$

${\lambda }_i$, the parameter of the Poisson distribution.

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

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

Returns

plekfloat, ndarray, shape $(max(\text{ll},\text{lk}))$

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

pgtkfloat, ndarray, shape $(max(\text{ll},\text{lk}))$

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

peqkfloat, ndarray, shape $(max(\text{ll},\text{lk}))$

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

ivalidint, ndarray, shape $(max(\text{ll},\text{lk}))$

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

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, ${\lambda }_i\le 0.0$.

$ivalid[i-1]=2$

On entry, $k_i<0$.

$ivalid[i-1]=3$

On entry, ${\lambda }_i>{10}^6$.

Raises

NagValueError

(errno $2$)

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

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

(errno $3$)

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

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

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $l$ 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 Poisson distribution with parameter ${\lambda }_i$ $(>0)$. Then

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

The mean and variance of each distribution are both equal to ${\lambda }_i$.

prob_poisson_vector computes, for given ${\lambda }_i$ and $k_i$ the probabilities: $Prob\{X_i\le k_i\}$, $Prob\{X_i>k_i\}$ and $Prob\{X_i=k_i\}$ using the algorithm described in Knüsel (1986).

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