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
|
![](I/ERMFormulaBookAppendixContinuous2_files/image001.png)
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Parameters
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= inverse
scale (i.e. rate) parameter ( )
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Domain
|
![](I/ERMFormulaBookAppendixContinuous2_files/image004.png)
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Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image005.png)
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Cumulative distribution
function
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![](I/ERMFormulaBookAppendixContinuous2_files/image006.png)
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Mean
|
![](I/ERMFormulaBookAppendixContinuous2_files/image007.png)
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Variance
|
![](I/ERMFormulaBookAppendixContinuous2_files/image008.png)
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Skewness
|
![](I/ERMFormulaBookAppendixContinuous2_files/image009.png)
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(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixContinuous2_files/image010.png)
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Characteristic function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image011.png)
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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
|
F distribution
|
Common notation
|
![](I/ERMFormulaBookAppendixContinuous2_files/image016.png)
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Parameters
|
= degrees
of freedom (first) (positive integer)
= degrees
of freedom (second) (positive integer)
|
Domain
|
![](I/ERMFormulaBookAppendixContinuous2_files/image004.png)
|
Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image019.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image020.png)
|
Mean
|
![](I/ERMFormulaBookAppendixContinuous2_files/image021.png)
|
Variance
|
![](I/ERMFormulaBookAppendixContinuous2_files/image022.png)
|
Skewness
|
![](I/ERMFormulaBookAppendixContinuous2_files/image023.png)
|
(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixContinuous2_files/image024.png)
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Characteristic function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image025.png)
Where is the
confluent hypergeometric function of the second kind
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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
![](I/ERMFormulaBookAppendixContinuous2_files/image029.png)
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
|
![](I/ERMFormulaBookAppendixContinuous2_files/image031.png)
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Parameters
|
= shape
parameter
= location
parameter
= scale
parameter
|
Domain
|
![](I/ERMFormulaBookAppendixContinuous2_files/image035.png)
|
Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image036.png)
where
![](I/ERMFormulaBookAppendixContinuous2_files/image037.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixContinuous2_files/image038.png)
|
Mean
|
![](I/ERMFormulaBookAppendixContinuous2_files/image039.png)
where is Euler’s
constant, i.e. ![](I/ERMFormulaBookAppendixContinuous2_files/image041.png)
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Variance
|
![](I/ERMFormulaBookAppendixContinuous2_files/image042.png)
Where ![](I/ERMFormulaBookAppendixContinuous2_files/image043.png)
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Skewness
|
![](I/ERMFormulaBookAppendixContinuous2_files/image044.png)
where is the
Riemann zeta function, i.e. .
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(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixContinuous2_files/image047.png)
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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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