# WNChygepdf

WNChygepdf returns Wallenius' non-central hypergeometric probability density values.

## Syntax

• Wpdf=WNChygepdf(X,N,K,M,W)example

## Description

This function is taken from the toolbox "Generation of Random Variates" (function wallen_pdf.m) created by Jim Huntley, that can be found at the Mathworks page:

FSDA uses the function only to demonstrate the coherence of the non-central hypergeometric distribution probability density values with samples extracted with FSDA function randsampleFS using option for weighted sampling without replacement.

To illustrate the meaning of Wallenius' function parameters, let's use the classical urn example, with $m_{1}$ red balls and $m_{2}$ white balls, totalling $M = m_{1}+m_{2}$ balls. $N$ balls are drawn at random from the urn one by one without replacement. Each red ball has the weight $\omega_{1}$, and each white ball has the weight $\omega_{2}$; the probability ratio of red over white balls is then given by $W = \omega_{1} / \omega_{2}$.

 Wpdf =WNChygepdf(X, N, K, M, W) Probability of getting 0 to p successes in p weighted drawns without replacement.

## Examples

expand all

### Probability of getting 0 to p successes in p weighted drawns without replacement.

Problem description.

%we have 500 balls in the urn
M  = 500;
%we extract 3 balls, one at a time, without replacement
N  = 3;
%initially, in the urn we have 250 red and 250 white balls
K  = M/2;
%red balls are ten times the white balls
W  = 10;
% We compute the probability of getting 0, 1, 2 or 3 red balls in drawing
% 3 balls without replacement.
for x = 0:N
wpdf(x+1) = WNChygepdf(x,N,K,M,W);
end
disp('We have an urn with 2 groups of balls;');
disp('There are 250 balls in each group;');
disp('But the probability of getting a ball of one type is 10 times that of the other type;');
disp('Then:');
disp('    the probability of getting 0, or 1, or 2, or 3 balls');
disp('    of the first type in 3 (weighted) drawns is respectively:');
disp(wpdf);
We have an urn with 2 groups of balls;
There are 250 balls in each group;
But the probability of getting a ball of one type is 10 times that of the other type;
Then:
the probability of getting 0, or 1, or 2, or 3 balls
of the first type in 3 (weighted) drawns is respectively:
0.0007    0.0225    0.2262    0.7505



## Input Arguments

### X — Number of red balls sampled. Scalar.

Data Types: single|double

### N — Total number of balls sampled. Scalar.

Data Types: single|double

### K — Initial number of red balls in the urn. Scalar.

Data Types: single|double

### M — Total number of balls in urn before sampling. Scalar.

Data Types: single|double

### W — Probability ratio of red over white balls. Scalar.

Data Types: single|double

## Output Arguments

### Wpdf —Wallenius' pdf values. Probability of drawing exactly X of a possible K items in N drawings without replacement from a group of M objects, when objects are from two weighted groups

Array of numel(x) values.

Data Types - single|double.

## References

Fog, A. (2008), Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution, "Communications in Statistics - Simulation and Computation", Vol. 37, pp. 258-273.