Statistics[Distributions]
StudentT
Student-t distribution
Calling Sequence
Parameters
Description
Examples
References
StudentT(nu)
StudentTDistribution(nu)
nu
-
degrees of freedom
The Student-t distribution is a continuous probability distribution with probability density function given by:
f⁡t=Γ⁡ν2+12π⁢ν⁢Γ⁡ν2⁢1+t2νν2+12
subject to the following conditions:
0<ν
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.
with(Statistics):
X := RandomVariable(StudentT(nu)):
PDF(X, u);
Γ⁡ν2+12π⁢ν⁢Γ⁡ν2⁢1+u2νν2+12
PDF(X, .5);
0.5641895835⁢Γ⁡0.5000000000⁢ν+0.5000000000ν⁢Γ⁡0.5000000000⁢ν⁢1.+0.25ν0.5000000000⁢ν+0.5000000000
Mean(X);
undefinedν≤10otherwise
Variance(X);
undefinedν≤2ν−2+νotherwise
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.
See Also
Statistics
Statistics[RandomVariable]
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