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 noncentral 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
FisherSnedecor distribution. It commonly arises in statistical tests linked
to analysis of variance.
If and are
independent random variables then
The Fdistribution 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 subfamilies 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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