The Chi-Square Distribution
Main Concept
The Chi-Square ( χ^{2}) distribution is the distribution of the sum of squares of a number of independent normal random variables. The Chi-Square distribution can be used to test for goodness of fit of observed sample data to theoretical models, as well as in estimating variances. It is also commonly used in the analysis of contingency tables.
The Chi-Square distribution depends on one parameter, ν, which is the number of degrees of freedom.
The probability density function of the Chi-Square distribution is given by:
${\chi}^{2}\left(\mathrm{\nu}\right)\=\frac{{x}^{\frac{\mathrm{\nu}}{2}-1}{\ⅇ}^{-\frac{x}{2}}}{{2}^{\frac{\mathrm{\nu}}{2}}\Gamma \left(\frac{\mathrm{\nu}}{2}\right)}$,
where the parameter ν is the number of degrees of freedom and Γ denotes the Gamma function.
The cumulative probability function is given by:
$1-\frac{\mathrm{\Gamma}\left(\frac{1}{2}\mathrm{\nu}\,\frac{1}{2}x\right)}{\mathrm{\Gamma}\left(\frac{1}{2}\mathrm{\nu}\right)}$
Properties
The Chi-Squared distribution has the following properties:
PDF
$\frac{{x}^{\frac{\mathrm{\nu}}{2}-1}{\ⅇ}^{-\frac{x}{2}}}{{2}^{\frac{\mathrm{\nu}}{2}}\Gamma \left(\frac{\mathrm{\nu}}{2}\right)}$
The probability density function.
CDF
The cumulative distribution function.
Mean μ
ν
The mean of the distribution μ is equal to the number of degrees of freedom, μ = ν .
Variance ${\mathrm{\sigma}}^{2}$
2ν
The variance, σ^{2}, is two times the number of degrees of freedom, σ^{2} = 2 ν .
Additionally:
When the number of degrees of freedom is greater or equal to 2, the maximum value occurs when ${\mathrm{\χ}}^{2}equals;\mathrm{nu;}-2$.
As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.
Change the value for the number of degrees of freedom n to see how the Chi-Squared Distribution changes:
ν =
μ_{x} =
σ^{2} =
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