What is the Meaning of Inverse CDF?

[scothoward]

Homework Statement

Hi,

There is one theorm in my Applied Probability course that I am having trouble understanding. It has to do with how to derive various types of random variables from the transformation X= g(U)

It says, Let U be a uniform (0,1) random variable and let F(x) denote a cumulative distribution function with an inverse F^-1(u) defined for 0<u<1. The random variable X = F^-1(U) has a CDF FX(x) = F(x).

I think my lack of understand of what F^-1(u) is hindering my understanding of this theorm. Would you be able to explain?

Homework Equations

X= g(U)

The Attempt at a Solution

From what I understand and the examples looked at, this theorm allows us
derive random variables of different types, using the uniform random
variable.

 

[lippyka]

F^-1(u) is the inverse of the cumulative distribution function. It is a function that takes the probability (u) and returns the corresponding value of the random variable (x). In other words, it is the distribution function for the random variable X. Therefore, the theorm states that if we take the uniform random variable U and use the inverse of the cumulative distribution function to transform it into a random variable X, then the CDF of X will be the same as the CDF of U.