Chi Square Random Variable

While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss chi square random variable prior.

What is chi-square random variable?

Let there be independent and identically distributed Gaussian random variables with mean and variance and we form a new random variable,

.

Then is a chi square random variable with degrees of freedom.

There are two types of chi square distribution. The first is obtained when has a zero mean and is called central chi square distribution. The second is obtained when has a non-zero mean and is called non-central chi square distribution. Four our discussion, we will focus only on central chi square distribution.

PDF of chi-square random variable with one degree of freedom

Using the text in Chapter 2 of [DIGITAL-COMMUNICATION: PROAKIS] as reference.

The most simple example of a chi square random variable is

,

where
is a Gaussian random variable with zero mean and variance .

The PDF of is
.

By definition, the cumulative distribution function (CDF) of is
.

This simplifies to

.

Differentiating the above equation with respect to to find the probability density function,

.

Summarizing, the pdf of chi square random variable with one degree of freedom is,

.

PDF of chi-square random variable with two degrees of freedom

Chi square random variable with 2 degrees of freedom is,

,

where,
and are independent Gaussian random variables with zero mean and variance .

In the post on Rayleigh random variable, we have shown that PDF of the random variable,

where is

.

For our current analysis, we know that

.

Differentiating both sides,

.

Applying this to the above equation, pdf of chi square random variable with two degrees of freedom is,
.

PDF of chi-square random variable with m degrees of freedom

The probability density function is,

, where

the Gamma function is defined as,

,

p an integer > 0

.

I do not know the proof for deriving the above equation. If any one of you know of good references, kindly let me know. Thanks. 🙂

Simulation Model

Just for your reference, Matlab/Octave simulation model performing the following is provided

(a) Generate chi square random variables having m=1, 2, 3, 4, 5 degrees of freedom

(b) Probability density function is computed and plotted

Click here to download: Matlab/Octave script for simulating PDF of chi square random variable

Figure: PDF of chi square random variable (=1)

Reference

[DIGITAL-COMMUNICATION: PROAKIS] Digital Communications, by John Proakis