The chi-square distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The ChiSquare variate with nu degrees of freedom is equivalent to the Gamma variate with scale $2$ and shape nu/2:  ChiSquare(nu) ~ Gamma(2,nu/2).

The ChiSquare variate is related to the FRatio variate by the formula FRatio(nu,omega) ~ (ChiSquare(nu)*omega)/(ChiSquare(omega)*nu)

The ChiSquare variate is related to the Normal variate and the StudentT variate by the formula StudentT(nu) ~ Normal(0,1)/sqrt(ChiSquare(nu)/nu)

Note that the ChiSquare command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{ChiSquare}\left(\mathrm{\nu }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{u^{\frac{\mathrm{\nu }}{2}-1}{\&ExponentialE;}^{-\frac{u}{2}}}{2^{\frac{\mathrm{\nu }}{2}}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{0.7788007831{0.5}^{0.5000000000\mathrm{\nu }-1.}}{{2.}^{0.5000000000\mathrm{\nu }}\mathrm{\Gamma }\left(0.5000000000\mathrm{\nu }\right)}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{\nu }$

$\mathrm{Variance}\left(X\right)$

$2\mathrm{\nu }$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol 1: Distribution Theory.