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Statistics[Distributions]

  

Gumbel

  

Gumbel distribution

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Gumbel(a, b)

GumbelDistribution(a, b)

Parameters

a

-

location parameter

b

-

scale parameter

Description

• 

The Gumbel distribution is a continuous probability distribution with probability density function given by:

ft=ⅇtabⅇⅇtabb

  

subject to the following conditions:

a::real&comma;0<b

• 

The Gumbel distribution is also known as the type I extreme value distribution.

• 

The Gumbel variate with location parameter a and scale parameter b is related to the Exponential variate with scale parameter b according to the formula:  Gumbel(a,b) ~ a - log(Exponential(b))

• 

Note that the Gumbel command is inert and should be used in combination with the RandomVariable command.

Examples

withStatistics&colon;

XRandomVariableGumbela&comma;b&colon;

PDFX&comma;u

&ExponentialE;uab&ExponentialE;&ExponentialE;uabb

(1)

PDFX&comma;0.5

&ExponentialE;1.0.51.ab&ExponentialE;1.&ExponentialE;1.0.51.abb

(2)

MeanX

b&gamma;&plus;a

(3)

VarianceX

16b2&pi;2

(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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