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 Vectorized Routines in the G01 Chapter Introduction for further information.

References

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

Parameters

Compulsory Input Parameters

1: $n (\ln)$ – int64int32nag_int array: $n_{i}$ , the first parameter of the binomial distribution with $n_{i} = n (j)$ , $j = ((i - 1) mod \ln) + 1$ , for $i = 1, 2, \dots, \max (\ln, lp, lk)$ .

Constraint: $n (j) \geq 0$ , for $j = 1, 2, \dots, \ln$ .
2: $p (lp)$ – double array: $p_{i}$ , the second parameter of the binomial distribution with $p_{i} = p (j)$ , $j = ((i - 1) mod lp) + 1$ .

Constraint: $0.0 < p (j) < 1.0$ , for $j = 1, 2, \dots, lp$ .
3: $k (lk)$ – int64int32nag_int array: $k_{i}$ , the integer which defines the required probabilities with $k_{i} = k (j)$ , $j = ((i - 1) mod lk) + 1$ .

Constraint: $0 \leq k_{i} \leq n_{i}$ .

Optional Input Parameters

1: $\ln$ – int64int32nag_int scalar: Default: the dimension of the array n.
The length of the array n

Constraint: $\ln > 0$ .
2: $lp$ – int64int32nag_int scalar: Default: the dimension of the array p.
The length of the array p

Constraint: $lp > 0$ .
3: $lk$ – int64int32nag_int scalar: Default: the dimension of the array k.
The length of the array k

Constraint: $lk > 0$ .

Output Parameters

1: $plek (:)$ – double array

The dimension of the array plek will be

\max (\ln, lp, lk)

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

, the lower tail probabilities.

2: $pgtk (:)$ – double array

The dimension of the array pgtk will be

\max (\ln, lp, lk)

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

, the upper tail probabilities.

3: $peqk (:)$ – double array

The dimension of the array peqk will be

\max (\ln, lp, lk)

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

, the point probabilities.

4: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (\ln, lp, lk)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

n_{i} < 0

$ivalid (i) = 2$

On entry,	$p_{i} \leq 0.0$ ,
or	$p_{i} \geq 1.0$ .

$ivalid (i) = 3$

On entry,	$k_{i} < 0$ ,
or	$k_{i} > n_{i}$ .

$ivalid (i) = 4$

On entry,

n_{i}

is too large to be represented exactly as a real number.

$ivalid (i) = 5$

On entry,

the variance (

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

) exceeds

10^{6}

5: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: On entry, at least one value of n, p or k was invalid.
Check ivalid for more information.

$ifail = 2$: Constraint: $\ln > 0$ .

$ifail = 3$: Constraint: $lp > 0$ .

$ifail = 4$: Constraint: $lk > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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).

Further Comments

The time taken by nag_stat_prob_binomial_vector (g01sj) 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.

Example

This example reads a vector of values for

n

p

and

k

, and prints the corresponding probabilities.

Open in the MATLAB editor: g01sj_example

function g01sj_example


fprintf('g01sj example results\n\n');

n = [int64(4); 19;     100;     2000];
p = [     0.500;  0.440;   0.750;    0.330];
k = [int64(2); 13;      67;      700];

[plek, pgtk, peqk, ivalid, ifail] = ...
  g01sj(n, p, k);

fprintf('    n     p      k     plek      pgtk      peqk\n');
ln  = numel(n);
lp  = numel(p);
lk  = numel(k);
len = max ([ln, lp, lk]);
for i=0:len-1
  fprintf('%5d%8.3f%5d%10.5f%10.5f%10.5f\n', n(mod(i,ln)+1), ...
          p(mod(i,lp)+1), k(mod(i,lk)+1), plek(i+1), pgtk(i+1), peqk(i+1));
end

g01sj example results

    n     p      k     plek      pgtk      peqk
    4   0.500    2   0.68750   0.31250   0.37500
   19   0.440   13   0.99138   0.00862   0.01939
  100   0.750   67   0.04460   0.95540   0.01700
 2000   0.330  700   0.97251   0.02749   0.00312

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox