Statistics and Probability with Applications for Engineers and Scientists, Preliminary Edition

Chapter 6

Distribution of Functions of Random Variables

The focus of this chapter is on the distributions of functions of random variables. The topics covered are:

• Joint distributions of two discrete random variables
• Joint distributions of two continuous random variables
• The mean value and variance of functions of two random variables
• Conditional distributions
• Correlation between two random variables
• Joint distributions of several random variables
• The moment generating function of two and more random variables

Learning Outcomes

After studying this chapter, the reader will be able to

• Understand the concept of discrete and continuous distributions of function of two or more random variables and use them to solve real-world problems.
• Understand the concept of marginal and conditional probability distributions.
• Determine the moment-generating functions of function of two or more random variables.

Introduction

In Chapters 4 and 5, we discussed various phenomena by enlisting a (single) random variable and studying its distribution and characteristics. However, in many situations, experiments are performed that involve two or more random variables. For example, we may be interested in the diameter and length of rods, the number of dots on two dice when rolled simultaneously, say (X, Y), where 1 ≤ X ≤ 6, 1 ≤ Y ≤ 6, or the composition of a monel (70 percent nickel, 30 percent copper) alloy, where we may focus on solid contents, say X, and liquid content Y, which again we would quote ...