Why does the random number generator in MATLAB fail a particular test of randomness?

The RAND function in MATLAB is not random in generating numbers that are small. The generation of very small values following runs of not-small values is particularly low. Here is an example that illustrates the problem:

rand('state',0);
n = 5*10^7;
delta = .05;
runs = diff(find(rand(n,1)<delta))-1;
y = histc(runs, 0:100) ./ length(runs);
plot(0:100,(1-delta).^(0:100).*delta,'k--', 0:100,y,'b-');
title('Distribution of run lengths')
xlabel('Run Length'); ylabel('Relative Frequency');
legend({'Expected (Geometric) Distribution' 'Actual Distribution'})

This code plots the number of random numbers that must be generated until another very small random number is generated. From the plot we can see that there is a noticeable decrease in runs of length 27

Answer 1

[MathWorks Support Team]

This enhancement has been incorporated in Release 14 Service Pack 3 (R14SP3). For previous product releases, read below for any possible workarounds:

This problem is due to some of the internal parameters used in the MATLAB random number generator. It is a shortcoming of the design of the algorithm, not a bug in the code. (As a note of interest, similar shortcomings are not uncommon in all pseudo-random number generators).

To work around this issue, you can download a different random number generator from the MATLAB Central File Exchange. It is called the Mersenne Twister, and is a relatively recent uniform random number generator algorithm. This is available at the following URL:

https://www.mathworks.com/matlabcentral/fileexchange/6614-mersenne-twister