Enterprise Risk Management Formula Book
Appendix A.2: Probability Distributions:
Continuous (univariate) distributions (a) Normal, uniform, chi-squared
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Distribution name
Normal distribution
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Common notation
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Parameters
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 = scale
parameter ( )
 = location
parameter
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Domain
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Probability density
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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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:
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Distribution name
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Uniform distribution
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Common notation
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Parameters
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 = boundary
parameters ( )
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Domain
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Probability density function
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Cumulative distribution function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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Its non-central moments ( are
 . Its
median is .
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Distribution name
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Chi-squared
distribution
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Common notation
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Parameters
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 = degrees
of freedom (positive integer)
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Domain
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Probability density
function
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Cumulative distribution
function
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Mean
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Variance
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Skewness
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(Excess) kurtosis
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Characteristic function
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Other comments
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Its median is approximately  . Its mode
is  . Is also
known as the central chi-squared distribution (when there is a need to
contrast it with the noncentral chi-squared distribution).
In the special case of  the
cumulative distribution function simplifies to  .
The chi-squared 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
chi-squared variables is also chi-squared 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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