How to efficiently generate a random integer within a range from an arbitrary probability distribution

I need to generate a random integer within a range from an arbitrary probability distribution, within a loop of 100000 iterations. My implementation works, but I am not sure it is mathematically clean, and it takes forever:

pdf = [ 0.9 0.3 0.003 0.1 0.07 0.0005 0.003 0.15 0.009 0.08 ]; % discrete prob distrib function
cdf = cumsum(pdf); % cumulative distribution function
cdf = cdf / max(cdf); cdf(1) = 0; % normalization
index = ceil(interp1(cdf, [1:numel(pdf)], rand(1)))

Notice that the pdf above is just an example: my actual case is a vector of about 500 numbers.

Here is a different solution, which seems mathematically cleaner, but does not work for my overall problem, and is just as slow as above:

pdf = [ 0.9 0.3 0.003 0.1 0.07 0.0005 0.003 0.15 0.009 0.08 ]; % discrete prob distrib function
cdf = cumsum(pdf); % cumulative distribution function
cdf = cdf - min(cdf); cdf = cdf / max(cdf); % normalization
index = round(interp1(cdf, [1:numel(pdf)], rand(1)))

Is there a more efficient/correct way to do this?

Answer 1

[the cyclist]

I think that

index = sum(rand()>cdf)+1;

will be much faster than using interp1 as you do, and will give the same result.

[the cyclist]

Also, I don't think your method of calculating the cdf is correct, because it will never select index=1, even though the pdf has support there.

To correct this, I think you want to use

cdf = [0 cdf];

in place of

cdf(1) = 0;

and modify my solution to just

index = sum(rand()>cdf);

[Paolo Binetti]

Your solution seems to work perfectly for my problem. It is about 7 times faster, which is a major improvement.

I have tried "randsample" from the Octave statistics package (should be more or less equivalent to the Matlab function) with the following syntax:

index = randsample(numel(pdf),1,true,pdf);

but it is even slower than my initial solution. Is this normal?

And you are totally right about the wrong way I used to to start cdf from zero. I had in mind your way, but did not use it because I thought it would imply never selecting the end index ... but that's not a problem with your implementation, right?

[the cyclist]

I double-checked by generating 100,000 samples, and the results seem right.

[the cyclist]

Regarding speed ...

I get roughly equivalent results between my solution and randsample. Note that both of these solutions can be vectorized:

index = sum(rand(N,1)>cdf,2)';

and

index = randsample(1:numel(pdf),N,true,pdf);

These will both generate N = 100,000 samples in a few milliseconds.

I did not try to vectorize your solution, but as it currently stands, it took 5 seconds to generate that many.

Answer 2

[the cyclist]

Do you have the Statistics and Machine Learning Toolbox? If so, you can do this with the randsample command.

[Paolo Binetti]

Thank you. No, I do not have this toolbox but I'll see if I can get it.