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### Enterprise Risk Management Formula Book Appendix A.2: Probability Distributions: Continuous (univariate) distributions (c) generalised Pareto, lognormal, Student’s t

Distribution name

Generalised Pareto distribution (GPD)

Common notation

Parameters

= shape parameter

= location parameter

= scale parameter ()

Domain

Probability density function

where

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

GPD is used in the peaks over thresholds variant of extreme value theory

 Distribution name Lognormal distribution Common notation Parameters = scale parameter ()  = location parameter Domain Probability density function Cumulative distribution function Mean Variance Skewness (Excess) kurtosis Characteristic function No simple expression that is not divergent Other comments The median of a lognormal distribution is  and its mode is .   The truncated moments of   are:

 Distribution name (Standard) Student’s t distribution Common notation Parameters = degrees of freedom (, usually  is an integer although in some situations a non-integral  can arise) Domain Probability density function Cumulative distribution function where Mean Variance Skewness (Excess) kurtosis Characteristic function where  is a Bessel function Other comments 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:   If  is even and  then , if  is odd and  then  and if  is odd and   then  is undefined.

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