O'Reilly logo

Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition by Kishor S. Trivedi

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 c03-math-0003 denote the class of measurable subsets of S. Now if X is to be a random variable, it is natural to require that c03-math-0006 be well defined for every real number x. In other words, if X is to be a random variable defined on a probability space c03-math-0009, we require that c03-math-0010 be an event (i.e., a member of ). We are, therefore, led to the following extension of our earlier ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required