Non-central Chi square distribution

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If squares of k independent standard normal random variables are added, it gives rise to central Chi-squared distribution with ‘k’ degrees of freedom. Instead, if squares of k independent normal random variables with non-zero means are added, it gives rise to non-central Chi-squared distribution. Non-central Chi-square distribution is related to Ricean distribution, whereas the central Chi-squared distribution is related to Rayleigh distribution.

The non-central Chi-squared distribution is a generalization of Chi-square distribution. A non-central Chi squared distribution is defined by two parameters: 1) degrees of freedom () and 2) non-centrality parameter .

As we know from previous article, the degrees of freedom specify the number of independent random variables we want to square and sum-up to make the Chi-squared distribution. Non-centrality parameter is the sum of squares of means of the each independent underlying normal random variable.

The non-centrality parameter is given by

The PDF of the non-central Chi-squared distribution having degrees of freedom and non-centrality parameter is given by

Here, the random variable is central Chi-squared distributed with degrees of freedom. The factor gives the probabilities of Poisson distribution. Thus, the PDF of the non-central Chi-squared distribution can be termed as the weighted sum of Chi-squared probabilities where the weights being equal to the probabilities of Poisson distribution.

Method of Generating non-central Chi-squared random variable:

The procedure for generating the samples from a non-central Chi-squared random variable is as follows.

● For a given degree of freedom , let the normal random variables be with variances and mean respectively.
● The goal is to add squares of these independent normal random variables with variances set to one and means satisfying the condition set by equation (1).
● Set and
● Generate standard normal random variables and one normal random variable with and
● Squaring and summing-up all the random variables gives the non-central Chi-squared random variable.
● The PDF of the generated samples can be plotted using the histogram method described here.

Matlab Code:

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

Python Code:

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

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

ks=np.asarray([2,4]) #degrees of freedoms to simulate
ldas = np.asarray([1,2,3]) #non-centrality parameters 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):
    for j,lda in enumerate(ldas):
        #Generate non-central Chi-squared distributed random numbers
        X = np.random.noncentral_chisquare(df=k, nonc = lda, size = nSamp)
        ax.hist(X,bins=500,density=True,label=r'$k$={} $\lambda$={}'.format(k,lda),\
        histtype='step',alpha=0.75, linewidth=3)

ax.set_xlim(left=0,right=30);ax.legend()
ax.set_title('PDFs of non-central Chi square distribution');
plt.show()

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For further reading

[1] David A. Harville, “Linear Models and the Relevant Distributions and Matrix Algebra”, 978-1138578333, Chapman and Hall/CRC, 1 edition, March 2018.↗

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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