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Statistics[Distributions]

  

ChiSquare

  

chi-square distribution

 

Calling Sequence

Parameters

Description

Examples

References

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:

ft=0t<0tν21&ExponentialE;t22ν2Γν2otherwise

  

subject to the following conditions:

0<ν

• 

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

withStatistics&colon;

XRandomVariableChiSquare&nu;&colon;

PDFX&comma;u

&lcub;0u<0u12&nu;1&ExponentialE;12u212&nu;&Gamma;12&nu;otherwise

(1)

PDFX&comma;0.5

0.77880078310.50.5000000000&nu;1.2.0.5000000000&nu;&Gamma;0.5000000000&nu;

(2)

MeanX

&nu;

(3)

VarianceX

2&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.

See Also

Statistics

Statistics[Distributions]

Statistics[RandomVariable]

 


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