Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition

Chapter 3Continuous Random Variables

3.1 Introduction

So far, we have considered discrete random variables and their distributions. In applications, such random variables denote the number of objects of a certain type, such as the number of job arrivals to a file server in one minute or the number of calls into a telephone exchange in one minute.

Many situations, both applied and theoretical, require the use of random variables that are “continuous” rather than discrete. As described in the last chapter, a random variable is a real-valued function on the sample space S. When the sample space S is nondenumerable (as mentioned in Section 1.7), not every subset of the sample space is an event that can be assigned a probability. As before, let denote the class of measurable subsets of S. Now if X is to be a random variable, it is natural to require that be well defined for every real number x. In other words, if X is to be a random variable defined on a probability space , we require that be an event (i.e., a member of ). We are, therefore, led to the following extension of our earlier ...