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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