Probability and Stochastic Processes

Chapter 6Tools to study Probability. Generating Function, Moment Generating Function, Characteristic Function

In this chapter we begin studying statistics which are defined as any one dimensional combination of random variables. In particular, the most important statistic is the average of random variables and by extension (because that is how we calculate the average) the sum of random variables. As we shall see below, calculating the distribution of sums of random variables is very complicated. However, the tools we introduced in this chapter will allow us to calculate the distribution of sums quite easily provided that the sum's components are independent. These tools are also the primary instrument used in Chapter 7 which studies what happens with statistics when the number of components goes to infinity. In practice, we cannot work with infinity; nevertheless, we can determine the direction a method is going if it continues the current sampling procedure and, most importantly, we may calculate how many items we need to have a good estimate of the limit.

6.1 Sums of Random Variables. Convolutions

Any sum has at least two components. We start by analyzing the distribution of a sum of just two variables.

Let be two random variables. Let be the distribution functions of X and ...