void	g01sjc (Integer ln, const Integer n[], Integer lp, const double p[], Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], NagError *fail)

The function may be called by the names: g01sjc, nag_stat_prob_binomial_vector or nag_prob_binomial_vector.

3 Description

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} \geq 0

and

0 < p_{i} < 1

). Then

Prob {X_{i} = k_{i}} = (\begin{array}{l} n_{i} \\ k_{i} \end{array}) {p_{i}}^{k_{i}} {(1 - p_{i})}^{n_{i} - k_{i}}, k_{i} = 0, 1, \dots, n_{i} .

The mean of the each distribution is given by

n_{i} p_{i}

and the variance by

n_{i} p_{i} (1 - p_{i})

g01sjc computes, for given

n_{i}

p_{i}

and

k_{i}

, the probabilities:

Prob {X_{i} \leq 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 Section 2.6 in the G01 Chapter Introduction for further information.

4 References

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

5 Arguments

1: $\ln$ – Integer Input

On entry: the length of the array n.

Constraint:

\ln > 0

2: $n [\ln]$ – const Integer Input

On entry:

n_{i}

, the first parameter of the binomial distribution with

n_{i} = n [j]

j = (i - 1) mod \ln

, for

i = 1, 2, \dots, \max (\ln, lp, lk)

Constraint:

n [j - 1] \geq 0

, for

j = 1, 2, \dots, \ln

3: $lp$ – Integer Input

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const double Input

On entry:

p_{i}

, the second parameter of the binomial distribution with

p_{i} = p [j]

j = (i - 1) mod lp

Constraint:

0.0 < p [j - 1] < 1.0

, for

j = 1, 2, \dots, lp

5: $lk$ – Integer Input

On entry: the length of the array k.

Constraint:

lk > 0

6: $k [lk]$ – const Integer Input

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = k [j]

j = (i - 1) mod lk

Constraint:

0 \leq k_{i} \leq n_{i}

7: $plek [\dim]$ – double Output

Note: the dimension, dim, of the array plek must be at least

\max (\ln, lp, lk)

On exit:

Prob {X_{i} \leq k_{i}}

, the lower tail probabilities.

8: $pgtk [\dim]$ – double Output

Note: the dimension, dim, of the array pgtk must be at least

\max (\ln, lp, lk)

On exit:

Prob {X_{i} > k_{i}}

, the upper tail probabilities.

9: $peqk [\dim]$ – double Output

Note: the dimension, dim, of the array peqk must be at least

\max (\ln, lp, lk)

On exit:

Prob {X_{i} = k_{i}}

, the point probabilities.

10: $ivalid [\dim]$ – Integer Output

Note: the dimension, dim, of the array ivalid must be at least

\max (\ln, lp, lk)

On exit:

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, $p_{i} \leq 0.0$ , or, $p_{i} \geq 1.0$ .
$ivalid [i - 1] = 3$: 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 $10^{6}$ .

11: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE: On entry, $array size = ⟨ value ⟩$ .
Constraint: $lk > 0$ .

On entry, $array size = ⟨ value ⟩$ .
Constraint: $\ln > 0$ .

On entry, $array size = ⟨ value ⟩$ .
Constraint: $lp > 0$ .
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of n, p or k was invalid.
Check ivalid for more information.

7 Accuracy

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).

8 Parallelism and Performance

g01sjc is not threaded in any implementation.

9 Further Comments

The time taken by g01sjc to calculate each probability depends on the variance (

= n_{i} p_{i} (1 - p_{i})

) and on

k_{i}

. For given variance, the time is greatest when

k_{i} \approx n_{i} p_{i}

(

= the mean

), and is then approximately proportional to the square-root of the variance.

10 Example

This example reads a vector of values for

n

p

and

k

, and prints the corresponding probabilities.

g01sj: FL CL CPP AD

NAG CL Interfaceg01sjc (prob_​binomial_​vector)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g01sjc (prob_binomial_vector)