I would not position random as replacing the more specific pseudorandom number generation functions. See the Alternative Functionality section on that documentation page. The random function allows you to switch from which distribution you're drawing at run-time by passing a different input argument into the function rather than modifying your code. If you know which distribution you want to draw from at write-time you could call the specific <distribution>rnd function or (for certain distributions) a function in MATLAB as opposed to Statistics and Machine Learning Toolbox (betarnd for the Beta distribution, unifrnd from Statistics and Machine Learning or rand from MATLAB for continuous uniform distribution, etc.)
You can't exactly specify the discrete uniform distribution on [0, 1] with just a call to random, but you can specify the discrete uniform distribution on [1, N] by specifying 'unid' as the distribution and N as the parameter.
x = random('unid', 2, [1 1e5]); % 1e5 uniform numbers that are either 1 or 2
figure
histogram(x)
title('Discrete uniform numbers either 1 or 2')
What happens if you subtract 1 from the output of that random call? That will give you discrete uniformly distributed numbers from the set [0, N-1].
figure
histogram(x-1)
title('Discrete uniform numbers either 0 or 1')
Compare with randi:
y = randi([0 1], 1, 1e5); % different set of random numbers either 0 or 1
figure
histogram(y)
title("Uniform random numbers either 0 or 1")
Those look pretty uniformly distributed to me. Note that the second and third histograms are not necessarily identical. If you flip a coin twice then flip it twice more, the results of the latter two flips may not be identical to the results of the first two flips.
