Probability Theory, 2nd Edition

4 Operations on random variables

4.1 Mappings of random variables

This section is devoted to the following problem: let $\mathit{X}:\mathrm{\Omega }\to \mathbb{R}$ be a random variable and let $\mathit{f}:\mathbb{R}\to \mathbb{R}$ be some function. Set $\mathit{Y}:=\mathit{f}\mathrm{(}\mathit{X}\mathrm{)}$, that is, for all $\mathit{\omega }\in \mathrm{\Omega }$ we have $\mathit{Y}\mathrm{(}\mathit{\omega }\mathrm{)}=\mathit{f}\mathrm{(}\mathit{X}\mathrm{(}\mathit{\omega }\mathrm{)}\mathrm{)}$. Suppose the distribution of X is known. Then the following task arises:

Determine the distribution of $\mathit{Y}=\mathit{f}\mathrm{(}\mathit{X}\mathrm{)}$ for a given function $\mathit{f}:\mathbb{R}\to \mathbb{R}$.

For example, if $\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}={\mathit{t}}^2$, and we know the distribution of X, then we ask for the probability distribution of ${\mathit{X}}^2$. Is it possible to compute this by easy methods?

At the moment it is not clear at all whether $\mathit{Y}=\mathit{f}\mathrm{(}\mathit{X}\mathrm{)}$ is a random variable. Only if this is valid, the probability distribution ${\mathbb{P}}_{\mathit{Y}}$ is well defined. For arbitrary functions f, this need not to be true, they have to ...