Studying distributions of random variables and their basic quantitative properties, such as expressions for moments, occupies a central role in both statistics and probability. It turns out that a function called the probability generating function is often a very useful mathematical tool in studying distributions of random variables. It is useful to derive formulas for moments and for the pmf of random variables that appear too complicated at first glance. In this chapter, we introduce the probability generating function, study some of its properties, and apply it to a selection of examples. The moment generating function, which is related to the probability generating function, is also extremely useful as a mathematical tool in numerous problems and is also introduced in this chapter. Both the generating function and the moment generating function should be primarily treated as useful tools. They help us solve important problems, and therefore they are useful as mathematical tools.
KeywordsGenerate Function Independent Random Variable Open Interval Mathematical Tool Power Series Expansion
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