Finite Discrete Distributions
For a discrete probability distribution with a finite number n of indices at which the probability mass function f takes non-zero values, the basic sampling algorithm is straightforward. The interval [0, 1) is divided in n intervals [0, f(1)), [f(1), f(1) + f(2)), ... The width of interval i equals the probability f(i). One draws a uniformly distributed pseudo-random number X, and searches for the index i of the corresponding interval. The so determined i will have the distribution f(i).
Formalizing this idea becomes easier by using the cumulative distribution function
It is convenient to set F(0) = 0. The n intervals are then simply [F(0), F(1)), [F(1), F(2)), ..., [F(n − 1), F(n)). The main computational task is then to determine i for which F(i − 1) ≤ X < F(i).
This can be done by different algorithms:
- Linear search, computational time linear in n.
- Binary search, computational time goes with log n.
- Indexed search, also called the cutpoint method.
- Alias method, computational time is constant, using some pre-computed tables.
- There are other methods that cost constant time.
Read more about this topic: Non-uniform Random Numbers
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