Gaussian Distribution

After being stupid (trying to be clever with the indices, but being too clever) I developed a dynamic programming solution to the peg network problem.  The figure on the left shows the what happens if we fit the results to a Gaussian distribution; we get excellent agreement, with small residuals.  However, the residuals are not randomly distributed about zero, which usually provides information of a systematic error.  I don't have an explanation for why those features showed up.

function [pdf grid data ]= DPSol(n)
% Making sure the input is odd; if it is not making it the next odd
if mod(n,2) == 0
    n = n +1;
end

% Setting up the the grid to be two larger than the probabilities
grid = gridSetup(n+2);

p = 0.5;
pdf = zeros(n,2*n+1);
mid = ceil(length(pdf(1,:))/2);

% Itterating down the "pegs"
pdf(1,mid) = 1.0;
for j = 2:n
    % Setting the pegs on that row
    for i=(mid-j+1):(mid+j-1)
        if(mod(j,2)==0)
            if( mod(i,2) == 1)
                pdf(j,i) = (pdf(j-1,i-1)+pdf(j-1,i+1))*rand;
            end
        else
            if( mod(i,2) == 0)
                pdf(j,i) = (pdf(j-1,i-1)+pdf(j-1,i+1))*rand;
            end
        end
    end
end

In the version I used to derive the graph I used a constant probability of 0.5, the following figure was produced using a random probability at each intersection.  You can see the initial spike at the first node of 1.0, and after that it all seems to die out as the ball progresses its way down the network.