I am trying to find a more "intuitive" understanding of the Probability Integral Transform (for the sake of better understanding Copula Models).
As far as I understand, the Probability Integral Transform is used for relating any continuous probability distribution to the uniform probability distribution. This transform states that the inverse of the cumulative probability distribution function of any probability distribution follows a uniform probability distribution.
For example, take the Normal Distribution (any probability distribution could have been chosen):
Then, take the Cumulative Normal Distribution:
The inverse of this Cumulative Distribution will follow a Uniform Probability Distribution:
I tried to show this using the R programming language.
Below is an Exponential Distribution converted into a Uniform Distribution:
x <- rexp(10000, 0.5) y <- 1 - exp(-0.5*x) hist(x) hist(y)
And here is a Uniform Distribution converted into an Exponential Distribution:
x <- runif(10000) y <- log(-x + 1) / (-1*0.5) hist(x) hist(y)
My Question: I have seen the mathematical proof of this Transform - but is there an "intuitive explanation" that explains why the Uniform Probability Distribution relates so many Probability Distributions together?