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
|
= 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
|
Uniform distribution
|
Common notation
|

|
Parameters
|
= boundary
parameters ( )
|
Domain
|

|
Probability density function
|

|
Cumulative distribution function
|

|
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
|
Chi-squared
distribution
|
Common notation
|

|
Parameters
|
= degrees
of freedom (positive integer)
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Domain
|

|
Probability density
function
|

|
Cumulative distribution
function
|

|
Mean
|

|
Variance
|

|
Skewness
|

|
(Excess) kurtosis
|

|
Characteristic function
|

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Other comments
|
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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