# probability integral transform

Let \(X\) be a continuous random variable with cumulative distribution function \(F_X\). Then, the random variable \(Y=F_X(X)\) is uniformly distributed.

## 1. Intuition

## 2. Story 1

Imagine the grade distribution for a class. The PDF can be crazy as it wants to be, but the percentiles will be uniformly distributed. That is, there will be the same amount of people in the top 5% as the middle 5% and the bottom 5%. You can imagine calling students randomly. Asking them what their grade was will recover the grade distribution. Asking them what percentile they are in will recover the uniform distribution.

## 3. Story 2

Let \(X\) have PDF \(f(x)\) and CDF \(F_X(x)\). Imagine using vertical lines on a graph to divide the PDF into bins of equal probability. Imagine drawing samples from \(F_X(X)\) and think about what bins your sample will come from. Some bins will be very narrow, because the probability density is high there. In these narrow bins, the CDF is increasing quickly. But we are also sampling more from these bins. Or to be truthful, we are sampling from the narrow bin as much as any wider bin. But the height of the bin is taller to compensate for the fact that \(F_X(x)\) is changing quickly there. The height of the "wider" bins can be lower, because \(F_X(x)\) is changing slowly there. In the end, these two effects result in a wash, and \(F_X(X)\) is uniformly distributed.