The chi-square distribution is commonly used in hypothesis testing, particularly the chi-squared test for goodness of fit.

The chi-square distribution uses the following parameter.

Parameter | Description | Support |
---|---|---|

ν | Degrees of freedom | ν is a nonnegative integer value |

The probability density function (pdf) is

$$y=f\left(x|\nu \right)=\frac{{x}^{\left(\nu -2\right)/2}{e}^{-x/2}}{{2}^{\frac{\nu}{2}}\Gamma \left(\nu /2\right)}$$

where Γ( · ) is the Gamma function, ν is the
degrees of freedom, and *x* ≥ `0`

.

The cumulative distribution function (cdf) is

$$p=F(x|\nu )={\displaystyle {\int}_{0}^{x}\frac{{t}^{(\nu -2)/2}{e}^{-t/2}}{{2}^{\nu /2}\Gamma (\nu /2)}dt}$$

where Γ( · ) is the Gamma
function, ν is the degrees of freedom, and *x* ≥ `0`

.

The mean is ν.

The variance is `2`

ν.

The *χ*^{2} distribution
is a special case of the gamma distribution where *b* =
2 in the equation for gamma distribution below.

$$y=f\left(x|a,b\right)=\frac{1}{{b}^{a}\Gamma \left(a\right)}{x}^{a-1}{e}^{\frac{x}{b}}$$

The *χ*^{2} distribution
gets special attention because of its importance in normal sampling
theory. If a set of *n* observations is normally
distributed with variance *σ*^{2},
and *s*^{2} is the sample
standard deviation, then

$$\frac{\left(n-1\right){s}^{2}}{{\sigma}^{2}}\sim {\chi}^{2}\left(n-1\right)$$

This relationship is used to calculate confidence intervals
for the estimate of the normal parameter *σ*^{2 } in
the function `normfit`

.

Compute the pdf of a chi-square distribution with 4 degrees of freedom.

x = 0:0.2:15; y = chi2pdf(x,4);

Plot the pdf.

figure; plot(x,y)

The chi-square distribution is skewed to the right, especially for few degrees of freedom.

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