Random variables of the form
appear repeatedly in probability theory and applications. We could in principle derive the probability model of Wn from the PMF or PDF of X1,…, Xn. However, in many practical applications, the nature of the analysis or the properties of the random variables allow us to apply techniques that are simpler than analyzing a general n-dimensional probability model. In Section 9.1 we consider applications in which our interest is confined to expected values related to Wn, rather than a complete model of Wn. Subsequent sections emphasize techniques that apply when X1, …, Xn are mutually independent. A useful way to analyze the sum of independent random variables is to transform the PDF or PMF of each random variable to a moment generating function.
The central limit theorem reveals a fascinating property of the sum of independent random variables. It states that the CDF of the sum converges to a Gaussian CDF as the number of terms grows without limit. This theorem allows us to use the properties of Gaussian random variables to obtain accurate estimates of probabilities associated with sums of other random variables. In many cases exact calculation of these probabilities is extremely difficult.
The expected value of a sum of any random variables is the sum of the expected values. The variance of the sum ...