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NAG Toolbox

NAG Toolbox: nag_stat_prob_binomial_vector (g01sj)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_stat_prob_binomial_vector (g01sj) returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

Syntax

[plek, pgtk, peqk, ivalid, ifail] = g01sj(n, p, k, 'ln', ln, 'lp', lp, 'lk', lk)
[plek, pgtk, peqk, ivalid, ifail] = nag_stat_prob_binomial_vector(n, p, k, 'ln', ln, 'lp', lp, 'lk', lk)

Description

Let X = Xi: i=1 , 2 ,, m  denote a vector of random variables each having a binomial distribution with parameters ni and pi (ni0 and 0<pi<1). Then
ProbXi=ki= ni ki piki1-pini-ki,  ki=0,1,,ni.  
The mean of the each distribution is given by nipi and the variance by nipi1-pi.
nag_stat_prob_binomial_vector (g01sj) computes, for given ni, pi and ki, the probabilities: ProbXiki, ProbXi>ki and ProbXi=ki 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 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:     nln int64int32nag_int array
ni, the first parameter of the binomial distribution with ni=nj, j=i-1 mod ln+1, for i=1,2,,maxln,lp,lk.
Constraint: nj0, for j=1,2,,ln.
2:     plp – double array
pi, the second parameter of the binomial distribution with pi=pj, j=i-1 mod lp+1.
Constraint: 0.0<pj<1.0, for j=1,2,,lp.
3:     klk int64int32nag_int array
ki, the integer which defines the required probabilities with ki=kj, j=i-1 mod lk+1.
Constraint: 0 ki ni .

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 maxln,lp,lk
Prob Xi ki , the lower tail probabilities.
2:     pgtk: – double array
The dimension of the array pgtk will be maxln,lp,lk
Prob Xi > ki , the upper tail probabilities.
3:     peqk: – double array
The dimension of the array peqk will be maxln,lp,lk
Prob Xi = ki , the point probabilities.
4:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxln,lp,lk
ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry,ni<0.
ivalidi=2
On entry,pi0.0,
orpi1.0.
ivalidi=3
On entry,ki<0,
orki>ni.
ivalidi=4
On entry,ni is too large to be represented exactly as a real number.
ivalidi=5
On entry,the variance (=nipi1-pi) exceeds 106.
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 (=nipi1-pi) and on ki. For given variance, the time is greatest when kinipi (=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.
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

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