Enterprise Risk Management Formula Book
Appendix A.2: Probability Distributions:
Continuous (univariate) distributions (a) Normal, uniform, chisquared
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Distribution name
Normal distribution
Common notation
Parameters
= scale
parameter ()
= location
parameter
Domain
Probability density
function
Cumulative distribution
function
Mean
Variance
Skewness
(Excess) kurtosis
Characteristic function
Other comments
The normal distribution is also called the Gaussian
distribution. The unit normal (or standard normal) distribution
is .
The inverse unit normal distribution function (i.e. its
quantile function) is commonly written (also in
some texts and the
unit normal density function is commonly written . is also
called the probit function.
The error function distribution is , where is
now an inverse scale parameter .
The median and mode of a normal distribution are .
The truncated first moments of are:
where and are the
pdf and cdf of the unit normal distribution respectively.
The mean excess function of a standard normal distribution
is thus
The central moments of the normal distribution are:
Distribution name

Uniform distribution

Common notation


Parameters

= boundary
parameters ()

Domain


Probability density function


Cumulative distribution function


Mean


Variance


Skewness


(Excess) kurtosis


Characteristic function


Other comments

Its noncentral moments ( are
. Its
median is .

Distribution name

Chisquared
distribution

Common notation


Parameters

= degrees
of freedom (positive integer)

Domain


Probability density
function


Cumulative distribution
function


Mean


Variance


Skewness


(Excess) kurtosis


Characteristic function


Other comments

Its median is approximately . Its mode
is . Is also
known as the central chisquared distribution (when there is a need to
contrast it with the noncentral chisquared distribution).
In the special case of the
cumulative distribution function simplifies to .
The chisquared distribution with degrees
of freedom is the distribution of a sum of the squares of independent
standard normal random variables. A consequence is that the sum of independent
chisquared variables is also chisquared distributed. It is widely used in
hypothesis testing, goodness of fit analysis or in constructing confidence
intervals. It is a special case of the gamma distribution.
As , and

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