Lesson 22: Functions of One Random Variable

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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.