STA305H1 Lecture Notes - Lecture 4: Probability-Generating Function, Moment-Generating Function, Central Limit Theorem
18 Jan 2018
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Document Summary
Demonstration of the central limit theorem for sums of independent but not identically distributed random variables. Suppose that x1, , xn are independent random variables and s = x1 + + xn. In this document, we will outline a method for computing exact the exact distribution of. S when the summands {xi} take values on a nite set of non-negative integers. We can then use these exact distribution to examine the adequacy of normal approximations (motivated by the central limit theorem) under various conditions. Suppose that x is a discrete random variable whose possible values are the integers 0, 1, 2, , and de ne the probability mass function f (x) = p (x = x) for x = 0, 1, 2, , . Then we can de ne the probability generating function1 p(t) = e(tx) = 1the probability generating function is related to the moment generating function m(t) = e[exp(tx)] of.
