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Student[Statistics]

 ChiSquareRandomVariable
 Chi-square random variable

 Calling Sequence ChiSquareRandomVariable(nu)

Parameters

 nu - first parameter

Description

 • The chi-square random variable is a continuous probability random variable 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)

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X≔\mathrm{ChiSquareRandomVariable}\left(\mathrm{ν}\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{{u}}^{\frac{{1}}{{2}}{}{\mathrm{ν}}{-}{1}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{u}}}{{{2}}^{\frac{{1}}{{2}}{}{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{ν}}\right)}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.7788007831}{}{{0.5}}^{{0.5000000000}{}{\mathrm{ν}}{-}{1.}}}{{{2.}}^{{0.5000000000}{}{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left({0.5000000000}{}{\mathrm{ν}}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{ν}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${2}{}{\mathrm{ν}}$ (4)
 > $Y≔\mathrm{ChiSquareRandomVariable}\left(3\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 ${{}\begin{array}{cc}{0}& {x}{<}{0}\\ {1}{-}\frac{{2}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{x}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}{x}}{+}\frac{{1}}{{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{erfc}}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{x}}\right)\right)}{\sqrt{{\mathrm{π}}}}& {\mathrm{otherwise}}\end{array}$ (5)
 > $\mathrm{CDF}\left(Y,6,\mathrm{output}=\mathrm{plot}\right)$

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.

Compatibility

 • The Student[Statistics][ChiSquareRandomVariable] command was introduced in Maple 18.