se variables. The most well known of these theorems is referred to as the central-limit theorem (CLT). The CLT states that if the variance of the initial distribution is finite, the limiting PDF is the Gaussian (normal) distribution. The CLT thus explains why the Gaussian (normal) distribution is found in SO many random processes: such processes usually stem from the additive effect of several independent or uncorrelated random variables, which individually obey any PDF type. A simplified formulation of the CLT is as follows. Let x be a random variable of a given parent distribution P parent (x). The parent distribution is characterized by a mean <x> = N and a finite variance o2. Let X1, X2, Xk be a series of k independent samples from this distribution. Define the sum Sk=x1+x2t,...,k. (2.23) Since the random variables (or samples) are independent, the mean and the variance of the sum in Eq. (2.23) are SK > = kN and of = ko2, respectively. The CLT simply states that the probability distribution of Sk asymptotically becomes Gaussian (normal) as the number of samples k increases, or k 8. A more general formulation of the CLT assumes a set of independent random variables X1X2,....XX. Each of these variables X has a different probability distribution p;(x). Each distribution pj(x) has a mean N and a variance o2 Define the sum Sk=X1+ = X2 +, , Xk. Since the variables X1X2,....XX , are independent, the mean and variance of the sum are <Sk>=N1 = + N2+,..., Nk and 02-02-02 +,...,02, respectively. Just as in the previous formulation, the CLT simply states that the probability distribution of the sum Sk is asymptotically Gaussian (normal). I will not expand on the formal proof of the CLT, which is beyond the scope of this book.2 However, for both clarification and fun purposes, the reader might check up this proof through some nicely illustrated online experiments using interactive Java 2 Formal demonstrations of the CLT can be found in many academic books and in some websites. See, for instance: htp://mathworld.wolfram.co