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. Formal statement For any random variable , the random variable has the same law as , where is the generalized inverse of the cumulative distribution function of and is uniform on . For continuous random variables, the inverse probability integral transform is indeed the inverse of the probability integral transform, which states that for a continuous random variable with cumulative distribution function , the random variable is uniform on . Intuition From , we want to generate with CDF We assume to be a continuous, strictly increasing function, which provides good intuition. We want to see if we can find some strictly monotone transformation , such that . We will have where the last step used that when is uniform on . So we got to be the inverse function of , or, equivalently Therefore, we can generate from The method The problem that the inverse transform sampling method solves is as follows: Let be a random variable whose distribution can be described by the cumulative distribution function . We want to generate values of which are distributed according to this distribution. The inverse transform sampling method works as follows: Generate a random number from the standard uniform distribution in the interval , i.e. from Find the generalized inverse of the desired CDF, i.e. . Compute . The computed random variable has distribution and thereby the same law as . Expressed differently, given a cumulative distribution function and a uniform variable , the random variable has the distribution . In the continuous case, a treatment of such inverse functions as objects satisfying differential equations can be given. Some such differential equations admit explicit power series solutions, despite their non-linearity. Examples As an example, suppose we have a random variable and a cumulative distribution function In order to perform an inversion we want to solve for From here we would perform steps one, two and three. As another example, we use the exponential distribution with for x ≥ 0 (and 0 otherwise). By solving y=F(x) we obtain the inverse function It means that if we draw some from a and compute This has exponential distribution. The