\problemdata       % Takes 5 *mandatory* arguments
  {nonlinear-variance}     % latex-friendly label for the prob.
  {variance}               % The topic(s) of the problem content.
  {Rosen 4.5.46}           % Source (if known)
  {S02 cp12W-2}            % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
\begin{problem}
Provide an example that shows that the variance of the sum of two
random variables is not necessarily equal to the sum of their
variances, when the random variables are not independent.

\solution{A dramatic example is to take $Y = -X$. Then the sum of the
two random variables is 0, so the variance of $X+Y$ is 0. But the sum
of the variances is $2\variance{X}$, since $Y$ has the same variance
as $X$. Another example: let $X$ be equal to 1 if a flipped coin is heads, 0 otherwise. Similarly, let $Y$ be 1 if the same coin comes up tails. Then $\variance{X}
= \variance{Y} = 1/4$, but $X+Y=1$, so $\variance{X+Y}=0$.
}
\end{problem}
