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Statistics[Distributions]

 StudentT
 Student-t distribution

 Calling Sequence StudentT(nu) StudentTDistribution(nu)

Parameters

 nu - degrees of freedom

Description

 • The Student-t distribution is a continuous probability distribution with probability density function given by:

$f\left(t\right)=\frac{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}+\frac{1}{2}\right)}{\sqrt{\mathrm{\pi }\mathrm{\nu }}\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right){\left(1+\frac{{t}^{2}}{\mathrm{\nu }}\right)}^{\frac{\mathrm{\nu }}{2}+\frac{1}{2}}}$

 subject to the following conditions:

$0<\mathrm{\nu }$

 • The StudentT variate is related to the Normal variate and the ChiSquare variate by the formula StudentT(nu) ~ Normal(0,1)/sqrt(ChiSquare(nu)/nu)
 • The StudentT variate with degrees of freedom 1 is related to the standard Cauchy variate by StudentT(1) ~ Cauchy(0,1).
 • Note that the StudentT command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{StudentT}\left(\mathrm{ν}\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}\right)}{\sqrt{{\mathrm{π}}{}{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{}{\mathrm{ν}}\right){}{\left({1}{+}\frac{{{u}}^{{2}}}{{\mathrm{ν}}}\right)}^{\frac{{1}}{{2}}{}{\mathrm{ν}}{+}\frac{{1}}{{2}}}}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.5641895835}{}{\mathrm{Γ}}{}\left({0.5000000000}{}{\mathrm{ν}}{+}{0.5000000000}\right)}{\sqrt{{\mathrm{ν}}}{}{\mathrm{Γ}}{}\left({0.5000000000}{}{\mathrm{ν}}\right){}{\left({1.}{+}\frac{{0.25}}{{\mathrm{ν}}}\right)}^{{0.5000000000}{}{\mathrm{ν}}{+}{0.5000000000}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${{}\begin{array}{cc}{\mathrm{undefined}}& {\mathrm{ν}}{\le }{1}\\ {0}& {\mathrm{otherwise}}\end{array}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{}\begin{array}{cc}{\mathrm{undefined}}& {\mathrm{ν}}{\le }{2}\\ \frac{{\mathrm{ν}}}{{-}{2}{+}{\mathrm{ν}}}& {\mathrm{otherwise}}\end{array}$ (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.