Inverse transform sampling method

The inverse transform sampling method is a method of sampling a number at random from any probability distribution, given its cumulative distribution function (cdf).

The problem that the inverse transform sampling method solves is as follows:

• Let X be a random variable whose distribution can be described by the cdf d(x).
• We want to generate values of x which are distributed according to this distribution.

Many programming languages have the ability to generate pseudo-randomnumbers which are effectively distributed according to the standard uniform distribution. If a random variable has that distribution, then the probability of its falling within any subinterval (a, b) of the interval from 0 to 1 is just the length b - a of that subinterval.

The inverse transform sampling method works as follows:

1. Generate a random number from the standard uniform distribution; call this u.
2. Compute the value for x which has the associated cdf value u; call this x_chosen.
3. Take x_chosen to be the random number drawn from the distribution described by d(x).

The following diagram may help the reader to visualise how the method works:

image:inverse_transform_sampling.png

Sampling using the inverse transform method

See also

The rejection sampling method.