Algebraic Manipulation of Random Variables (Numerical)

Numerically approximate the PDF of a combination of the random variables X and Y.

% [z zC Z ZC] = transRV(jointXY,C,opp,n) Numerically approximate the PDF of
% a nonlinear combination of the random variables X and Y given the joint
% distribution of X and Y. The joint distribution (jointXY) should be
% generated using [jointXY, C] = hist3([x(:) y(:)]).
%
% z = transRV(jointXY,C,opp,n) Use the opperation opp which can
% be: '+', '-', '*', or '/', to generate a PDF z of length n.
%
% z = transRV(...,'ZLim',[minz maxz]) specify the limits on z.
%
% z = transRV(...,'Visual') display the area of integration and CDF of Z.
%
% [z zC Z ZC zlim] = transRV(...) returns the PDF z with values centered at
% zC, and the CDF Z with values centered at ZC. Also returns the limits on
% z.
%
% NOTE: Results may be noisy if the output vector z has high resolution
% (larger n) than the input distribution.
%
% EXAMPLE
%
% % Set the number of bins
% n = 200;
%
% % Generate random variables
% x = 2*randn(1000000,1);
% y = .5*randn(1000000,1);
%
% % Display joint distribution
% [jointXY C] = hist3([x y],[n n]);
% figure; imagesc(C{1},C{2},jointXY');
% set(gca,'YDir','normal')
%
% % Approximate new distribution
% [z zC Z ZC zlim] = transRV(jointXY,C,'*',n,'ZLim',[-5,5],'Visual');
%
% % Compare to the distribution of data under the same opperation
% zdata = x.*y;
% [N xc] = hist(zdata(zdata >= zlim(1) & zdata <= zlim(2)),n);
% bin_width = xc(2)-xc(1);
% N = N/sum(N.*bin_width);
%
% figure; bar(xc,N,'hist')
% hold on; plot(zC,z,'r')
% legend('Dist of Data','Approximation')

% Created by Aaron Swan - Summer 2010
% MERS Lab - Brigham Young University