Overview

In this chapter we introduce the cumulative distribution function (cdf) of a random variable X. The cdf is defined by ^[a]

As we shall see, knowing the cdf is equivalent to knowing the density or pmf of a random variable. By this we mean that if you know the cdf, then you can find the density or pmf, and if you know the density or pmf, then you can find the cdf. This is the same sense in which knowing the characteristic function is equivalent to knowing the density or pmf. Similarly, just as some problems are more easily solved using characteristic functions instead of densities, there are some problems that are more easily solved using cdfs instead of densities.

This chapter emphasizes three applications in which cdfs figure prominently: ( i) Finding the probability density of Y = g( X) when the function g and the density of X are given; ( ii) The central limit theorem; and ( iii) Reliability.

The first application concerns what happens when the input of a system g is modeled as a random variable. The system output Y = g( X) is another random variable, and we would like to compute probabilities involving Y. For example, g could be an amplifier, and we might need to find the probability that the output exceeds some danger level. If we knew the probability mass function or the density...