APPENDIX D
Summary of Common Random Variables
This appendix provides a quick reference of some of the most common random variables. Special functions that are used in this appendix are defined in the following list:
Continuous Random Variables
Arcsine
For any b > 0
(D.1)
(D.2)
(D.3)
Beta
For any a > 0 and b > 0
(D.4)
(D.5)
(D.6)
Cauchy
For any b > 0
(D.7)
(D.8)
(D.9)
Notes:
Chi-Square
For integer n > 0
(D.10)
(D.11)
(D.12)
(D.13)
Notes:
1) The Chi-Square random variable is a special case of the Gamma random variable.
2) The parameter n is referred to as the number of degrees of freedom of the chi-square random variable.
3) The Chi-Square random variable is formed by a transformation of the form 
, where the Zk
are IID zero-mean, unit variance, Gaussian random variables.
Erlang
For any integer n > 0 and any b > 0
(D.14)
(D.15)
(D.16)
(D.17)
Exponential
For any b > 0
(D.19)
(D.20)
(D.21)
(D.22)
Notes:
F
For any integers n > 0 and m > 0
(D.24)
(D.25)
Note:
Gamma
For any a > 0 and b > 0
(D.26)
(D.27)
(D.28)
(D.29)
Note:
Gaussian
For any μ and any σ > 0
(D.30)
(D.31)
(D.32)
(D.33)
Gaussian-Multivariate
For any n element column vector μ and any valid n × n covariance matrix c
(D.34)
(D.35)
(D.36)
Laplace
For any b > 0
(D.37)
(D.38)
(D.39)
(D.40)
Log-Normal
For any μ and any σ> 0
(D.41)
(D.42)
(D.43)
1) The log-normal random variable is formed by a transformation of the form X = exp (Z), where Z is a Gaussian random variable with mean μ and variance σ2.
2) It is common to find instances in the literature where σ is referred to as the standard deviation of the log-normal random variable. This is a misnomer. The quantity σ is not the standard deviation of the log-normal random variable but rather is the standard deviation of the underlying Gaussian random variable.
Nakagami
For any b > 0 and m > 0
(D.44)
(D.45)
(D.46)
Rayleigh
For any σ > 0
(D.47)
(D.48)
(D.49)
1) The Rayleigh random variable arises when performing a Cartesian to Polar transformation of two independent zero-mean Gaussian random variables. That is, if Y1 and Y2 are independent zero-mean Gaussian random variables with variances of σ2, then 
follows a Rayleigh distribution.
2) The Rayleigh random variable is a special case of the Rician random variable.
Rician
For any a ≥ 0 and any σ > 0
(D.50)
(D.51)
(D.52)
(D.53)
Notes:
1) The Rician random variable arises when performing a Cartesian to Polar transformation of two independent Gaussian random variables. That is, if Y1 and Y2 are independent Gaussian random variables with means of μ1 and μ2, respectively and equal variances of σ2, then 
follows a Rician distribution, with 
.
2) The ratio a2/σ2 is often referred to as the Rician parameter or the Rice factor. As the Rice factor goes to zero, the Rician random variable becomes a Rayleigh random variable.
Student t
For any integer n > 0
(D.54)
(D.55)
1) This distribution was first published by W. S. Gosset in 1908 under the pseudonym “Student.” Hence this distribution has come to be known as the Student’s t-distribution.
2) The parameter n is referred to as the number of degrees of freedom.
3) If Xi, i = 1, 2, …, n, is a sequence of IID Gaussian random variables and 
and 
are the sample mean and sample variance, respectively, then the ratio 
will have a t-distribution with n − 1 degrees of freedom.
Uniform
For any a ≤ b
(D.56)
(D.57)
(D.58)
(D.59)
Discrete Random Variables
Bernoulli
For 0 ≤ p ≤ 1
(D.63)
(D.64)
(D.65)
Binomial
For 0 ≤p ≤1 and any integer n > 0
(D.66)
(D.67)
(D.68)
Note:
Geometric
For 0 ≤ p ≤1
(D.69)
(D.70)
(D.71)
Pascal (or Negative Binomial)
For 0 ≤ q ≤ 1 and any integer n > 0
(D.72)
(D.73)
(D.74)
Poisson
For any b > 0
(D.75)
(D.76)
(D.77)