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 (
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
Here, the random variable
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
● The goal is to add squares of these
● Set
● Generate
● Squaring and summing-up all the
● 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
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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 Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |  Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. |
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