# Lesson 22: Functions of One Random Variable

### Introduction

We'll begin our exploration of the distributions of functions of random variables, by focusing on simple functions of *one* random variable. For example, if *X *is a continuous random variable, and we take a function of *X*, say:

*Y *= *u*(*X*)

then* Y *is also a continuous random variable that has its own probability distribution. We'll learn how to find the probability density function of *Y,* using two different techniques, namely the **distribution function technique** and the **change-of-variable technique**. At first, we'll focus only on one-to-one functions. Then, once we have that mastered, we'll learn how to modify the change-of-variable technique to find the probability of a random variable that is derived from a two-to-one function. Finally, we'll learn how the inverse of a cumulative distribution function can help us simulate random numbers that follow a particular probability distribution.

### Objectives

- To learn how to use the distribution function technique to find the probability distribution of
*Y*=*U*(*X*), a one-to-one transformation of a random variable*X*. - To learn how to use the change-of-variable technique to find the probability distribution of
*Y*=*U*(*X*), a one-to-one transformation of a random variable*X*. - To learn how to use the change-of-variable technique to find the probability distribution of
*Y*=*U*(*X*), a two-to-one transformation of a random variable*X*. - To learn how to use a cumulative distribution function to simulate random numbers that follow a particular probability distribution.
- To understand all of the proofs in the lesson.
- To be able to apply the methods learned in the lesson to new problems.