Enterprise Risk Management Formula Book
Appendix A.2: Probability Distributions:
Continuous (univariate) distributions (c) generalised Pareto, lognormal,
Student’s t
[this page  pdf]
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
Other comments
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 nonintegral 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 noncentral moments if is
even and are:
If is even
and then , if is
odd and then and if is
odd and then is undefined.

NAVIGATION LINKS
Contents  Prev  Next