Weighted random numbers in MATLAB

Question

How to randomly pick up N numbers from a vector a with weight assigned to each number?

Let's say:

a = 1:3; % possible numbers
weight = [0.3 0.1 0.2]; % corresponding weights

In this case probability to pick up 1 should be 3 times higher than to pick up 2.

Sum of all weights can be anything.

Answer 1

R = randsample([1 2 3], N, true, [0.3 0.1 0.2])

randsample is included in the Statistics Toolbox

---

Otherwise you can use some kind of roulette-wheel selection process. See this similar question (although not MATLAB specific). Here's my one-line implementation:

a = 1:3;             %# possible numbers
w = [0.3 0.1 0.2];   %# corresponding weights
N = 10;              %# how many numbers to generate

R = a( sum( bsxfun(@ge, rand(N,1), cumsum(w./sum(w))), 2) + 1 )

Explanation:

Consider the interval [0,1]. We assign for each element in the list (1:3) a sub-interval of length proportionate to the weight of each element; therefore 1 get and interval of length 0.3/(0.3+0.1+0.2), same for the others.

Now if we generate a random number with uniform distribution over [0,1], then any number in [0,1] has an equal probability of being picked, thus the sub-intervals' lengths determine the probability of the random number falling in each interval.

This matches what I'm doing above: pick a number X~U[0,1] (more like N numbers), then find which interval it falls into in a vectorized way..

---

You can check the results of the two techniques above by generating a large enough sequence N=1000:

>> tabulate( R )
  Value    Count   Percent
      1      511     51.10%
      2      160     16.00%
      3      329     32.90%

which more or less match the normalized weights w./sum(w) [0.5 0.16667 0.33333]