tailieunhanh - Introduction to Probability - Chapter 10

Chapter 10 Generating Functions Generating Functions for Discrete Distributions So far we have considered in detail only the two most important attributes of a random variable, namely, the mean and the variance. We have seen how these attributes enter into the fundamental limit theorems of probability | Chapter 10 Generating Functions Generating Functions for Discrete Distributions So far we have considered in detail only the two most important attributes of a random variable namely the mean and the variance. We have seen how these attributes enter into the fundamental limit theorems of probability as well as into all sorts of practical calculations. We have seen that the mean and variance of a random variable contain important information about the random variable or more precisely about the distribution function of that variable. Now we shall see that the mean and variance do not contain all the available information about the density function of a random variable. To begin with it is easy to give examples of different distribution functions which have the same mean and the same variance. For instance suppose X and Y are random variables with distributions 1 2 3 4 5 6 PX 0 1 4 1 2 0 0 1 4 1234 5 6 PY V1 4 0 0 1 2 1 4 0 J Then with these choices we have E X E Y 7 2 and V X V Y 9 4 and yet certainly pX and pY are quite different density functions. This raises a question If X is a random variable with range xY x2 . of at most countable size and distribution function p pX and if we know its mean p E X and its variance a2 V X then what else do we need to know to determine p completely Moments A nice answer to this question at least in the case that X has finite range can be given in terms of the moments of X which are numbers defined as follows 365 366 CHAPTER 10. GENERATING FUNCTIONS 1k kth moment of X E X k xj kp xj j 1 provided the sum converges. Here p xj P X Xj . In terms of these moments the mean i and variance a2 of X are given simply by 1 11 2 2 a 12 - 11 so that a knowledge of the first two moments of X gives us its mean and variance. But a knowledge of all the moments of X determines its distribution function p completely. Moment Generating Functions To see how this comes about we introduce a new variable t and define a function g t as follows g t E .