Tutorial IV: Some Useful Probability Distribution

Binomial Distribution:

Let X be a discrete random variable that has two possible values says X = 1 or X = 0 with probabilities of p and 1-p respectively. Now suppose, we define Y as

pdf of Y will be

and CDF is

the above equation characterizes a binomially distributed random variable.

The first two moments of Y are :

E(Y) = np

E(Y2) = np(1-p) +n2p2

σ2 = np(1-p)

Uniform Distribution:

first two moments are:

E(X) = ½ (a + b)

E(X2)= 1/3(a2+b2+ab)

σ2 = 1/12 (a-b)2

Gaussian (normal) distribution:

PDF

CDF is

and erf(x) denotes the error function, defined as

in terms of erfc(x) CDF is given as

where erfc(x) = 1 – erf(x)

"The sum of n statistically independent Gaussian random variables is also Gaussian random variable"

Chi- Square (gamma) Distribution (with n degrees of freedom):

central chi square distribution

PDF

Cdf

First two moments are

E(Y) = nσ2

E(Y2) = 2nσ4 + n2σ4

σ2y = 2nσ4

for non central chi square distribution refer page nujmber 42 – 44 of Digital Communications 4/e John G. Proakis

Rayleigh Distribution

This is one of the most important probability distribution in Communicaiton Theory. This distribution is frequently used to model the statistics of signals transmitted through radio channels such as cellular radio. This distribution is closely related to the central chi-square distribution.

Let Y=X12 +X22 where X1 and X2 are zero mean statistically independent Gaussian random variables, each having a variance σ2 . It follows U is chi square distributed with two degrees of freedom. That means PDF of Y is

now, let define a new Random variable R which is

PDF of R is

CDF is

and moments are

E(Rk) = (2σ2)k/2Γ(1 + ½ k)

and the variance is

σ2r = (2 – ½ π)σ2

Lognormal Distribution:

a random variable X is normally distributed with mean m and variance σ2. Let us define a new random variable R which is related to X through X = ln R . Then the pdf of R is

The lognormal distribution is suitable for modeling the effect of shadowing of the signal due to large obstructions, such as tall building, in mobile radio communications.

Matlab Functions and Codes:

Distribution	Matlab Function
Erf, erfc	Erfc, erf, etc
Binomial	Binornd, binopdf	Statistical Toolbox -> Binomial Distribution
Gaussian (normal)	Normrnd, normfit	Statistical Toolbox -> Normal Distribution
Uniform	Unicdf,	Usually first generate random numbers according to distribution and then use respective funstions to genrate pdf, cdf etc. Like chi2pdf etc.
Chi Square	Chi2rnd, chi2pdf etc.
Rayliegh	Raylrnd, raylpdf etc.