Probability: Chi-Squared Distribution

Chi-Squared Distribution

Explained:

central chi-squared distribution

chi-squared distribution

degrees of freedom

non-central chi-squared distribution

non-centrality parameter

The chi-squared distribution arises frequently in applications because of its close association with the normal distribution.

Suppose Z is a standard normal random variable. How is distributed? The answer is a chi-squared . More generally, let be ν independent standard normal random variables, and let be constants. Then the random variable:

[1]

has a chi-squared distribution with ν degrees of freedom and non-centrality parameter

[2]


Ads by Contingency Analysis	Advertise on this site
<![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=84506;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=84506"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=84506;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]>

This result generalizes still further. It can be shown that any quadratic polynomial of a joint normal random vector can be expressed as a linear polynomial of independent chi-squared and normal random variables. Why is this result important? One reason is that it arises in value-at-risk (VaR) calculations, facilitating quadratic VaR measures (also called delta-gamma VaR measures).

We denote a chi-squared distribution that has ν degrees of freedom and non-centrality parameter with the notation: . If the is said to be centrally chi-squared. Otherwise, it is said to be non-centrally chi-squared. The probability density function (PDF) for a central chi-squared is

[3]

For the non-central chi-squared , it generalizes to

[4]

where Γ denotes the gamma function

[5]

The PDF of a is illustrated in Exhibit 1:

χ²(ν,δ²) Probability Density Function
Exhibit 1

PDF of a χ²(4,4) distribution.

The expectation, standard deviation, skewness and kurtosis of a random variable are:

Ads by Contingency Analysis

<![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82402;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82402"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82402;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]>

	[6]

	[7]

	[8]

	[9]

One caveat: Treatment of the non-centrality parameter is not standardized in the literature. Some authors define the parameter as in [2] but denote it simply δ. Others define the parameter differently, for example, taking a square root in [2] or dividing the sum by 2.