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
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Common notation
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Parameters
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 = inverse
scale (i.e. rate) parameter ( )
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Domain
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Probability density
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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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  .
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Distribution name
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F distribution
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Common notation
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Parameters
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 = degrees
of freedom (first) (positive integer)
 = degrees
of freedom (second) (positive integer)
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Domain
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Probability density
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Where  is the
confluent hypergeometric function of the second kind
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Other comments
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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.
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Distribution name
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Generalised extreme
value (GEV) distribution (for maxima)
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Common notation
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Parameters
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 = shape
parameter
 = location
parameter
 = scale
parameter
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Domain
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Probability density
function
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where
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Cumulative distribution
function
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Mean
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where  is Euler’s
constant, i.e. 
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Variance
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Where 
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Skewness
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where  is the
Riemann zeta function, i.e.  .
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(Excess) kurtosis
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Other comments
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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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