

\begin{slide}
\subsection{\bf 1.4\hspace{1em} Functions of a Random Variable}
\begin{itemize}
\item If $x$ is a random variable, then so is $y=f(x)$.  

\item Let $x$ have probability distribution $p(x)$ and $y$ probability
distribution $q(y)$.

\item Then the probability of $x$ lying in $[x, x+ dx]$ is $p(x) dx$:

\item Let $[y, y+dy]$ be associated interval in $y$.

\item Clearly, $p(x)dx = q(y)dy$ and 

\item $\frac{dy}{dx} = |f^\prime (x)|$, 

\item so
\begin{equation}
p(x) = q\left( f(x)\right) | f^\prime (x) |
\end{equation}

\item There are some obvious modifications of this if the
$x\leftrightarrow y$ relation is not single-valued.
\end{itemize}
\end{slide}

