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Statistics[Distributions]

 ChiSquare
 chi-square distribution

 Calling Sequence ChiSquare(nu) ChiSquareDistribution(nu)

Parameters

 nu - first parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{t}^{\frac{\mathrm{\nu }}{2}-1}{ⅇ}^{-\frac{t}{2}}}{{2}^{\frac{\mathrm{\nu }}{2}}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0<\mathrm{\nu }$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{ChiSquare}\left(\mathrm{\nu }\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{{u}}^{\frac{{\mathrm{\nu }}}{{2}}{-}{1}}{}{{ⅇ}}^{{-}\frac{{u}}{{2}}}}{{{2}}^{\frac{{\mathrm{\nu }}}{{2}}}{}{\mathrm{\Gamma }}{}\left(\frac{{\mathrm{\nu }}}{{2}}\right)}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.7788007831}{}{{0.5}}^{{0.5000000000}{}{\mathrm{\nu }}{-}{1.}}}{{{2.}}^{{0.5000000000}{}{\mathrm{\nu }}}{}{\mathrm{\Gamma }}{}\left({0.5000000000}{}{\mathrm{\nu }}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{\nu }}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${2}{}{\mathrm{\nu }}$ (4)

References

 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.