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Statistics[Distributions]

  

StudentT

  

Student-t distribution

 

Calling Sequence

Parameters

Description

Examples

References

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:

ft=Γν2+12πνΓν21+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.

Examples

withStatistics&colon;

XRandomVariableStudentT&nu;&colon;

PDFX&comma;u

&Gamma;12&nu;&plus;12&pi;&nu;&Gamma;12&nu;1&plus;u2&nu;12&nu;&plus;12

(1)

PDFX&comma;0.5

0.5641895835&Gamma;0.5000000000&nu;&plus;0.5000000000&nu;&Gamma;0.5000000000&nu;1.&plus;0.25&nu;0.5000000000&nu;&plus;0.5000000000

(2)

MeanX

&lcub;undefined&nu;10otherwise

(3)

VarianceX

&lcub;undefined&nu;2&nu;2&plus;&nu;otherwise

(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.

See Also

Statistics

Statistics[Distributions]

Statistics[RandomVariable]