Enterprise Risk Management Formula Book
Appendix A.4: Probability Distributions: Location
and scale adjusted distributions
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The location
and scale of any probability distribution can be adjusted by using the
(linear) transform where and
are
constants. This leaves the skew and (excess kurtosis) unaltered but alters the
mean and variance as follows: and .
In some cases, the typical distributional specification
already includes such components. For example, the normal distribution is the
location and scale adjusted version of the unit normal distribution .
In other cases, the standard distributional specification
does not include such adjustments. For example, the (standard) Student’s t
distribution depends one just one parameter, its degrees of freedom. In these
cases, the distributional definition noted above may need to be expanded to
include location and/or scale adjusted variants when fitting data to such
distributions.
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