Linear transformation of random variables.

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VI.

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e introduce the notation MATH to describe the statement "random variable has distribution $g\left( y\right)$ ".

Proposition

Let be a vector of random variables and MATH for some function . Set for some deterministic square matrix and vector . If $\det A\neq0$ then MATH

Proof

For any domain of the -space we can write MATH MATH We make the change of variables in the last integral.

(Linear transformation of random variables)

Hence

The linear transformation $\sigma\xi+\mu$ is distributed as . The $\xi$ was defined in the section ( Definition of normal variable ).

For two independent standard normal variables (s.n.v.) $\xi_{1}$ and $\xi _{2},$ the combination is distributed as .

A product of normal variables is not a normal variable. See the section ( Chi squared distribution ).