Enterprise Risk Management Formula Book
Appendix A.2: Probability Distributions:
Continuous (univariate) distributions (c) generalised Pareto, lognormal,
Student’s t
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Distribution name
Generalised
Pareto distribution (GPD)
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Common notation
|
![](I/ERMFormulaBookAppendixContinuous3_files/image001.png)
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Parameters
|
= shape
parameter
= location
parameter
= scale
parameter ( )
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Domain
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![](I/ERMFormulaBookAppendixContinuous3_files/image006.png)
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Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image007.png)
where
![](I/ERMFormulaBookAppendixContinuous3_files/image008.png)
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Cumulative distribution
function
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![](I/ERMFormulaBookAppendixContinuous3_files/image009.png)
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Mean
|
![](I/ERMFormulaBookAppendixContinuous3_files/image010.png)
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Variance
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![](I/ERMFormulaBookAppendixContinuous3_files/image011.png)
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Skewness
|
![](I/ERMFormulaBookAppendixContinuous3_files/image012.png)
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(Excess) kurtosis
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![](I/ERMFormulaBookAppendixContinuous3_files/image013.png)
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Other comments
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GPD is used in the peaks over thresholds variant of
extreme value theory
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Distribution name
|
Lognormal
distribution
|
Common notation
|
![](I/ERMFormulaBookAppendixContinuous3_files/image014.png)
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Parameters
|
= scale
parameter ( )
= location
parameter
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Domain
|
![](I/ERMFormulaBookAppendixContinuous3_files/image015.png)
|
Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image016.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image017.png)
|
Mean
|
![](I/ERMFormulaBookAppendixContinuous3_files/image018.png)
|
Variance
|
![](I/ERMFormulaBookAppendixContinuous3_files/image019.png)
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Skewness
|
![](I/ERMFormulaBookAppendixContinuous3_files/image020.png)
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(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixContinuous3_files/image021.png)
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Characteristic function
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No simple expression that is not divergent
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Other comments
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The median of a lognormal distribution is and its
mode is .
The truncated moments of are:
![](I/ERMFormulaBookAppendixContinuous3_files/image025.png)
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Distribution name
|
(Standard)
Student’s t distribution
|
Common notation
|
![](I/ERMFormulaBookAppendixContinuous3_files/image026.png)
|
Parameters
|
= degrees
of freedom ( , usually is
an integer although in some situations a non-integral can
arise)
|
Domain
|
![](I/ERMFormulaBookAppendixContinuous3_files/image029.png)
|
Probability density
function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image030.png)
|
Cumulative distribution
function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image031.png)
where ![](I/ERMFormulaBookAppendixContinuous3_files/image032.png)
|
Mean
|
![](I/ERMFormulaBookAppendixContinuous3_files/image033.png)
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Variance
|
![](I/ERMFormulaBookAppendixContinuous3_files/image034.png)
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Skewness
|
![](I/ERMFormulaBookAppendixContinuous3_files/image035.png)
|
(Excess) kurtosis
|
![](I/ERMFormulaBookAppendixContinuous3_files/image036.png)
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Characteristic function
|
![](I/ERMFormulaBookAppendixContinuous3_files/image037.png)
where is a
Bessel function
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Other comments
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The Student’s t distribution (more simply the t
distribution) arises when estimating the mean of a normally distributed
population when sample sizes are small and the population standard deviation
is unknown.
It is a special case of the generalised hyperbolic
distribution.
Its non-central moments if is
even and are:
![](I/ERMFormulaBookAppendixContinuous3_files/image041.png)
If is even
and then , if is
odd and then and if is
odd and then is undefined.
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