In Statistics, there is a topic called Sampling Distributions, where the goal is to find the distribution of a function g(X) of a random variable X, given the distribution of X. The following problem is a simple illustration of the topic.
Problem: Given that X follows a continuous uniform distribution distribution over the closed interval [0,1], we are asked to find out the distribution of the square of X. In other words, if X~Uniform[0,1], then what is the distribution of g(X)=X².
We shall explain the solution to the problem in a few steps:
Step 1: We first write down the probability density function (PDF) of X.
Step 2: Define a new random variable Y:=g(X)=X².
Step 3: Compute the cumulative distribution function (CDF) of Y.
Step 4: Differentiate the CDF of Y to obtain the PDF of Y.
Takeaway from the problem: It turns out that if X~Uniform[0,1], then X²~Beta(1/2,1).
See the image below for a complete solution to the problem:
This is a completely wrong definition!
Clean derivation and a great visual reminder of how transformations affect probability density functions—love seeing this broken down step by step