Chi square distribution – demystified

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A random variable is always associated with a probability distribution. When the random variable undergoes mathematical transformation the underlying probability distribution no longer remains the same. Consider a random variable whose probability distribution function (PDF) is a standard normal distribution ( and ). Now, if the random variable is squared (a mathematical transformation), then the PDF of is no longer a standard normal distribution. The new transformed distribution is called Chi square Distribution with degree of freedom. The PDF of and are plotted in Figure 1.

The mean of the random variable is and for the transformed variable Z², the mean is given by . Similarly, the variance of the random variable is , whereas the variance of the transformed random variable is . In addition to the mean and variance, the shape of the distribution is also changed. The distribution of the transformed variable is no longer symmetric. In fact, the distribution is skewed to one side. Also the random variable can take only positive values whereas the random variable can take negative values too (note the x-axis in the plots above).

Since the new transformation is based on only one parameter (), the degree of freedom for this transformation is . Therefore, the transformed random variable follows – “Chi-square distribution with degree of freedom”.
Suppose, if are independent random variables that follows standard normal distribution( and ), then the transformation,

is a Chi square distribution with k degrees of freedom. The following figure illustrates how the definition of the Chi square distribution as a transformation of normal distribution for degree of freedom and degrees of freedom. In the same manner, the transformation can be extended to degrees of freedom.

The above equation is derived from random variables that follow standard normal distribution. For a standard normal distribution, the mean . Therefore, the transformation is called central Chi-square distribution. If, the underlying random variables follow normal distribution with non-zero mean, then the transformation is called non-central Chi-square distribution [2] . In channel modeling, the central Chi-squared distribution is related to Rayleigh Fading scenario and the non-central Chi-square distribution is related to Rician Fading scenario.

Mathematically, the PDF of the central Chi-squared distribution with degrees of freedom is given by

The mean and variance of the central Chi-squared distributed random variable is given by

Relation to Rayleigh distribution

The connection between Chi square distribution and the Rayleigh distribution can be established as follows

1. If a random variable has standard Rayleigh distribution, then the transformation follows chi-square distribution with degrees of freedom.
2. If a random variable has the chi-square distribution with degrees of freedom, then the transformation has standard Rayleigh distribution.

Applications:

Chi-square distribution is used in hypothesis testing (to compare the observed data with expected data that follows a specific hypothesis) and in estimating variances of a parameter.

Matlab Simulation:

Check this book for full Matlab code.
Wireless Communication Systems using Matlab – by Mathuranathan Viswanathan

Python Code

Python numpy package has a chisquare() generator, which can be used in a straightforward manner to obtain the Chi square distributed sequences.

#---------Chi square distribution gaussianwaves.com-----
import numpy as np
import matplotlib.pyplot as plt
#%matplotlib inline
plt.style.use('ggplot')

ks=np.arange(start=1,stop=6,step=1) #degrees of freedoms to simulate
nSamp=1000000 #number of samples to generate

fig, ax = plt.subplots(ncols=1, nrows=1, constrained_layout=True)

for i,k in enumerate(ks):
    #Generate central Chi-square distributed random numbers
    X = np.random.chisquare(df=k, size = nSamp)
    ax.hist(X,bins=500,density=True,label=r'$k$={}'.format(k), \
    histtype='step',alpha=0.75, linewidth=3)

ax.set_xlim(left=0,right=8);ax.set_ylim(bottom=0,top=0.5);ax.legend();
ax.set_title('PDFs of Chi square distribution');
ax.set_xlabel(r'$\chi_k^2$');ax.set_ylabel(r'$f_{\chi_k^2}(x)$');
plt.show()

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For further reading

[1] Ernie Croot, “Notes on Chi-squared distribution”, Georgia institute of technology, School of mathematics, Oct 2005.↗

Similar topics

Random Variables - Simulating Probabilistic Systems ● Introduction ● Plotting the estimated PDF ● Univariate random variables  □ Uniform random variable  □ Bernoulli random variable  □ Binomial random variable  □ Exponential random variable  □ Poisson process  □ Gaussian random variable  □ Chi-squared random variable  □ Non-central Chi-Squared random variable  □ Chi distributed random variable  □ Rayleigh random variable  □ Ricean random variable  □ Nakagami-m distributed random variable ● Central limit theorem - a demonstration ● Generating correlated random variables  □ Generating two sequences of correlated random variables  □ Generating multiple sequences of correlated random variables using Cholesky decomposition ● Generating correlated Gaussian sequences  □ Spectral factorization method  □ Auto-Regressive (AR) model

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