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Enterprise Risk Management Formula Book

Appendix A.2: Probability Distributions: Continuous (univariate) distributions (b) exponential, F, generalised extreme value (GEV) (and Frechét, Gumbel and Weibull)

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Distribution name

Exponential distribution

Common notation

Parameters

 = inverse scale (i.e. rate) parameter ()

Domain

Probability density function

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Characteristic function

Other comments

Also called the negative exponential distribution. The mode of an exponential distribution is 0. The exponential distribution describes the time between events if these events follow a Poisson process. It is not the same as the exponential family of distributions. The quantile function, i.e. the inverse cumulative distribution function, is .

 

The non-central moments ( are . Its median is .

 

 

Distribution name

F distribution

Common notation

Parameters

 = degrees of freedom (first) (positive integer)

 = degrees of freedom (second) (positive integer)

Domain

Probability density function

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Characteristic function

Where  is the confluent hypergeometric function of the second kind

Other comments

The F distribution is a special case of the Pearson type 6 distribution. It is also known as Snedecor’s F or the Fisher-Snedecor distribution. It commonly arises in statistical tests linked to analysis of variance.

 

If  and  are independent random variables then

 

 

The F-distribution is a particular example of the beta prime distribution.

 

The mode is . There is no simple closed form for the median.

 

 

Distribution name

Generalised extreme value (GEV) distribution (for maxima)

Common notation

Parameters

 = shape parameter

 = location parameter

 = scale parameter

Domain

Probability density function

where

Cumulative distribution function

Mean

where  is Euler’s constant, i.e.

Variance

Where

Skewness

where  is the Riemann zeta function, i.e. .

(Excess) kurtosis

Other comments

 defines the tail behaviour of the distribution. The sub-families defined by  (Type I),  (Type II) and  (Type III) correspond to the Gumbel, Frechét and Weibull families respectively.

 

An important special case when analysing threshold exceedances involves  (and normally ) and this special case may be referred to as .

 


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