Non-central Chi-squared Distribution

If squares of k independent standard normal random variables (mean=0, variance=1) 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 mean (mean \( \neq \) 0 , variance=1) are added, it gives rise to non-central Chi-squared 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 is given by

In the above equation, \( f_{\chi_k}(x;\lambda)\) indicates the non-central Chi-squared distribution with \(k\) degrees of freedom with non-centrality parameter specified by \( \lambda\) and the factor \( f_{\chi_{k+2n}}(x) \) indicates the ordinary central Chi-squared distribution with \(k+2n\) degrees of freedom.

The factor \( e^{-\frac{\lambda}{2}}\frac{{(\frac{\lambda}{2})}^n}{n!}\) gives the probabilities of Poisson Distribution. So, the PDF of the non-central Chi-squared distribution can be termed as a weighted sum of Chi-squared probability with weights being equal to the probabilities of Poisson distribution.

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

Parameters required: k – the degrees of freedom and \( \lambda \) – non-centrality parameter.

  • For a given degree of freedom (k), let the k normal random variables be \( X_1,X_2,…,X_k \) with variances \( {\sigma_1}^2 ,{\sigma_2}^2,…,{\sigma_k}^2 =1 \) and mean \( \mu_1,\mu_2,…\mu_k\)
  • Now, our goal is to add squares of the k independent normal random variables with variances=1 and means satisfying the following criteria
  • Put \( \mu_1=\sqrt{\lambda} \) and \( \mu_2,…\mu_k = 0\)
  • Generate k-1 standard normal random variables with \( \mu=0 \) and \( {\sigma}^2 =1\) and one normal random variable with \( \mu= \sqrt{\lambda} \) and \( {\sigma}^2 =1\)
  • Squaring and summing-up all the k random variables give the non-central Chi-squared random variable
  • The PDF can be plotted using histogram method

Non-central Chi-squared distribution is related to Rician Distribution and the central Chi-squared distribution is related to Rayleigh distribution.

Matlab Code:

Check this book for full Matlab code.
Simulation of Digital Communication Systems Using Matlab – by Mathuranathan Viswanathan

Simulated output - non-central Chi squared Distribution with k degrees of freedom

See Also:

[1] Introduction to Random Variables, PDF and CDF
[2] Central Chi-squared distribution and its simulation in Matlab
[3] Uniform Random Variables and Uniform Distribution
[4] Derivation of Error Rate Performance of an optimum BPSK receiver in AWGN channel
[5] Eb/N0 Vs BER for BPSK over Rician Fading Channel
[6] BER Vs Eb/N0 for QPSK modulation over AWGN
[7] BER Vs Eb/N0 for 8-PSK modulation over AWGN
[8] Simulation of M-PSK modulation techniques in AWGN channel
[9] Performance comparison of Digital Modulation techniques
[10]How to use Histogram function in Matlab to plot the estimated PDF curve

External Links:

[1] Chi-Square Test Penn State University
[2] Java Applet – Chi Square goodness of Fit test – created by David Eck and modified by Jim Ryan – Mathbeans project
[3] Chi-Square Test for variance ,e-handbook of statistical methods,National Institute of Standards and Technology
[4] Dr. Claude Moore,Estimation of Variance,Cape Fear Community College

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