# Chi-Squared Distribution

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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 $$Z$$ whose probability distribution function (PDF) is a standard normal distribution ($$\mu=0$$ and $$\sigma^2=1$$). Now, if the random variable is squared (a mathematical transformation), then the PDF of $$Z^2$$ is no longer a standard normal distribution. The new transformed distribution is called Chi-Squared Distribution with $$1$$ degree of freedom. The PDF of $$Z$$ and $$Z^2$$ are plotted in Figure 1.

The mean of the random variable $$Z$$ is $$E(Z) = 0$$ and for the transformed variable Z2, the mean is given by $$E(Z^2)=1$$. Similarly, the variance of the random variable $$Z$$ is $$\sigma^2_Z=1$$, whereas the variance of the transformed random variable $$Z^2$$ is $$\sigma^2_{Z^2}=2$$. In addition to the mean and variance, the shape of the distribution is also changed. The distribution of the transformed variable $$Z^2$$ is no longer symmetric. In fact, the distribution is skewed to one side. Also the random variable $$Z^2$$ can take only positive values whereas the random variable $$Z$$ can take negative values too (note the x-axis in the plots above).

Since the new transformation is based on only one parameter ($$Z$$), the degree of freedom for this transformation is $$1$$. Therefore, the transformed random variable $$Z^2$$ follows – “Chi-squared distribution with $$1$$ degree of freedom”.
Suppose, if $$Z_1,Z_2,\cdots,Z_k$$ are independent random variables that follows standard normal distribution($$\mu=0$$ and $$\sigma^2=1$$), then the transformation,

$$\chi_k^2 = Z_1^2 + Z_2^2+ \cdots+Z_k^2$$

is a Chi-squared distribution with k degrees of freedom. The following figure illustrates how the definition of the Chi-squared distribution as a transformation of normal distribution for $$1$$ degree of freedom and $$2$$ degrees of freedom. In the same manner, the transformation can be extended to $$k$$ degrees of freedom.

The above equation is derived from $$k$$ random variables that follow standard normal distribution. For a standard normal distribution, the mean $$\mu=0$$. Therefore, the transformation $$\chi_k^2$$ is called $$central$$ Chi-square distribution. If, the underlying $$k$$ random variables follow normal distribution with non-zero mean, then the transformation $$\chi_k^2$$ is called $$non-central$$ Chi-square distribution [2] . In channel modeling, the central Chi-square 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 $$k$$ degrees of freedom is given by

$$f_{\chi_k^2 }(x)= \frac{1}{2^{\frac{k}{2}}\Gamma \left(\frac{k}{2}\right )}x^{\frac{k}{2}-1}e^{-\frac{x}{2}}$$

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

$$\mu = E\left[\chi_k^2\right] = k$$

$$\sigma^2 = var\left[\chi_k^2\right] = 2k$$

## Relation to Rayleigh distribution

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

1. If a random variable $$R$$ has standard Rayleigh distribution, then the transformation $$R^2$$ follows chi-square distribution with $$2$$ degrees of freedom.
2. If a random variable $$C$$ has the chi-square distribution with $$2$$ degrees of freedom, then the transformation $$\sqrt{C}$$ has standard Rayleigh distribution.

## Applications:

Chi-squared 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:

The following Matlab code is used to simulated central Chi-squared distributed variables with degrees of freedom = $$1,2,3,4$$ and $$5$$. Finally the PDFs of the simulated variables are plotted.

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