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Statistics[Distributions]

  

Logistic

  

logistic distribution

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Logistic(a, b)

LogisticDistribution(a, b)

Parameters

a

-

location parameter

b

-

scale parameter

Description

• 

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

ft=ⅇtabb1+ⅇtab2

  

subject to the following conditions:

a::real&comma;0<b

• 

The logistic variate Logistic(a,b) is related to the standardized variate Logistic(0,1) by Logistic(0,1) ~ (Logistic(a,b)-a)/b.

• 

The standard logistic variate is related to the standard Exponential variate by Logistic(0,1)  -log(exp(-Exponential(1))/(1+exp(-Exponential(1)))).

• 

The logistic variate with location parameter 0 and scale parameter b is related to two independent Gumbel variates G1 and G2 by Logistic(0,b) ~ G1 - G2.

• 

The standardized logistic variate is related to the Pareto variate with location parameter a and shape parameter c by Logistic(0,1)  logParetoa&comma;cac1.

• 

The standardized logistic variate is related to the Power variate with scale parameter 1 and shape parameter c by Logistic(0,1) ~ -log(Power(1,c)^(-c)-1).

• 

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

Examples

withStatistics&colon;

XRandomVariableLogistica&comma;b&colon;

PDFX&comma;u

&ExponentialE;uabb1&plus;&ExponentialE;uab2

(1)

PDFX&comma;0.5

&ExponentialE;0.51.abb1.&plus;&ExponentialE;0.51.ab2

(2)

MeanX

a

(3)

VarianceX

13b2&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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