Inverse transform sampling

Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function.

Inverse transformation sampling takes uniform samples of a number ${\displaystyle u}$ between 0 and 1, interpreted as a probability, and then returns the smallest number ${\displaystyle x\in \mathbb {R} }$ such that ${\displaystyle F(x)\geq u}$ for the cumulative distribution function ${\displaystyle F}$ of a random variable. For example, imagine that ${\displaystyle F}$ is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution.

Transformation from uniform sample to normal

${\displaystyle u}$	${\displaystyle F^{-1}(u)}$
.5	0
.975	1.95996
.995	2.5758
.999999	4.75342
1-2−52	8.12589

We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs to the left of that number. Intuitively, we are unlikely to choose a number in the far end of tails because there is very little area in them which would require choosing a number very close to zero or one.

Computationally, this method involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function (CDF) of the distribution (which maps a number in the domain to a probability between 0 and 1) and then inverting that function. This is the source of the term "inverse" or "inversion" in most of the names for this method. Note that for a discrete distribution, computing the CDF is not in general too difficult: we simply add up the individual probabilities for the various points of the distribution. For a continuous distribution, however, we need to integrate the probability density function (PDF) of the distribution, which is impossible to do analytically for most distributions (including the normal distribution). As a result, this method may be computationally inefficient for many distributions and other methods are preferred; however, it is a useful method for building more generally applicable samplers such as those based on rejection sampling.

For the normal distribution, the lack of an analytical expression for the corresponding quantile function means that other methods (e.g. the Box–Muller transform) may be preferred computationally. It is often the case that, even for simple distributions, the inverse transform sampling method can be improved on:^[1] see, for example, the ziggurat algorithm and rejection sampling. On the other hand, it is possible to approximate the quantile function of the normal distribution extremely accurately using moderate-degree polynomials, and in fact the method of doing this is fast enough that inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.^[2]