multcov(n,p)

function [C] = multcov(n,p)
% MULTCOV Covariance of a multinomial distribution.
%  [C] = MULTCOV(N,P) returns the covariance of a multinomial distribution 
%  with parameters N and P.
%  The Coariance is: 
%      Covariance = -N  Pi  Pj;  i,j = 1,2, ... ,k. 
%
%      Covariance = [  np1(1 - p1)    -np1p2          -np1pk
%                        -np1p2     np2(1 - p2)       -np2pk
%                           .            .      .           .
%                           .            .        .         .
%                           .            .          .       .
%                        -np1pk       -np2pk        npk(1 - pk)  ].
%
%  Syntax: function [C] = multcov(n,p) 
%      
%  Inputs:
%       n - number of trials.
%       p - vector of associated probabilities. 
%  Outputs:
%       C - multinomial covariance matrix.
%
%  Example from the Dr. Hossein Arsham Statistic Site (http://www.staff.vu.edu.au/
%  sarath/Business-stats/opre504.htm). For a multinomial distribution function
%  (http://www.staff.vu.edu.au/sarath/Business-stats/opre504.htm#rmultinomial). 
%  Consider two investment alternatives, Investment I and Investment II with the
%  characteristics outlined in the following table: 
%
%                - Two Investments -  
%           Investment I  Investment II 
%           Payoff  Prob.  Payoff  Prob. 
%              1    0.25      3    0.33 
%              7    0.50      5    0.33 
%             12    0.25      8    0.34 
%
%  For the investment I we are interested to get the multinomial covariance.
%
%  Calling on Matlab the function: 
%             [C] = multcov(n,p)
%
%  where n = 20  and  p = [.25,.5,.25]
%
%  Answer is:
%
%  C =
%      1.7452   -1.5706   -0.1745
%     -1.5706    1.7452   -0.1745
%     -0.1745   -0.1745    0.3490
%
%  Created by A. Trujillo-Ortiz, R. Hernandez-Walls and A. Castro-Perez
%             Facultad de Ciencias Marinas
%             Universidad Autonoma de Baja California
%             Apdo. Postal 453
%             Ensenada, Baja California
%             Mexico.
%             [e-mail]
%  Copyright (C) April 05, 2005.
%
%  To cite this file, this would be an appropriate format:
%  Trujillo-Ortiz, A., R. Hernandez-Walls and A. Castro-Perez. (2005). multcov:
%    Covariance of a multinomial distribution. A MATLAB file. [WWW document]. URL 
%    http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=7375
%
%  References:
% 
%  Arsham, H. Statistic Site http://www.staff.vu.edu.au/sarath/Business-stats/opre504.htm
%          http://www.staff.vu.edu.au/sarath/Business-stats/opre504.htm#rmultinomial
%

if nargin < 2, 
   error('You need to input two arguments.');
   return,
end;

k = length(p);

pp = sum(p);

if pp ~= 1,
    error('The sum of the input probabilities must be equal 1.')
    return,
end;

for i = 1:k,
    for j = 1:k,
        if (i==j);
            C(i,j) = n*p(i)*(1.0-p(i));
        else
            C(i,j) = -n*p(i)*p(j);
        end;
    end;
end;

return,