Distributions derived from the normal distribution

Abstract

In Sections 4.2.4 and 4.3.1 we defined the normal (or Gaussian) distributions for both single and multiple variables and discussed their properties. The normal distribution plays a central role in the mathematical theory of statistics for at least two reasons. First, the normal distribution often describes a variety of physical quantities observed in the real world. In a communication system, for example, a received waveform is often a superposition of a desired signal waveform and (unwanted) noise process, and the amplitude of the noise is often normally distributed, because the source of such noise is usually what is known as thermal noise at the receiver front. The normality of thermal noise is a good example of manifestation in the real world of the CLT, which says that the sum of a large number of independent RVs, properly scaled, tends to be normally distributed. In Chapter 3 we saw that the binomial distribution and the Poisson distribution also tend to a normal distribution in the limit. We also discussed the CLT and asymptotic normality. The second reason for the frequent use of the normal distribution is its mathematical tractability. For instance, sums of independent normal RVs are themselves normally distributed. Such reproductivity of the distribution is enjoyed only by a limited class of distributions (that is, binomial, gamma, Poisson). Many important results in the theory of statistics are founded on the assumption of a normal distribution.