# APPENDIX

# A Advanced Probabilistic Models

In the study of digital communications presented in preceding chapters, the Gaussian, Rayleigh, and Rician distributions featured in the formulation of probabilistic models in varying degrees. In this appendix we describe three relatively advanced distributions:

- the chi distribution;
- the log-normal distribution;
- the Nakagami distribution.

The chi distribution is featured in the study of diversity-on-receive techniques in Chapter 9 on signaling across fading channels. Just as importantly, the log-normal distribution was mentioned in passing in the context of shadowing in wireless communications, also in
Chapter 9. The Nakagami distribution is the most advanced of all the three:

- it includes the Rayleigh distribution as a special case;
- its shape is similar to the Rician distribution;
- it is flexible in its applicability.

## A.1 The Chi-Square Distribution

A *chi-square χ*^{2} distributed random variable is produced, for example, when a Gaussian random variable is passed through a squaring device. Viewed in this manner, there are two kinds of *χ*^{2} distributions:

1. * Central χ*^{2} * distribution*, which is produced when the Gaussian random variable has zero mean.

2. *Noncentral χ*^{2} *distribution*, which is produced when the Gaussian random variable has a nonzero mean.

In this appendix, we will discuss only the central form of the distribution.

Consider, then, a standard Gaussian random variable *X*, which has zero mean and unit variance, as shown by

Let the ...