/

### 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.