<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{2800hw}
\usepackage{amsmath,amssymb,latexsym}
\usepackage{utf8}

\topic{Random variables}
\homework{4}

\begin{document}
\maketitle

\begin{exercises}
\item Consider the following experiment.  To decide where to travel to, a
traveler throws a dart at a map of a region.  The region contains four states,
named $a$, $b$, $c$, and $d$.  Assume that the probability of hitting a given
state is proportional to the area of the state, and that the probability of
hitting one of the four states is one (that is, the traveler never misses the map).

Let $X$ be the variable giving the cost of a ticket to the selected state, and
let $Y$ by the variable giving the population of the selected state.  Facts about
the states are contained in the following table:

\begin{center}
\begin{tabular}{llrr}
State &amp; Area     &amp; Cost     &amp; Population   \\ \hline
$a$ &amp; $1000~\textrm{km}^2$ &amp; $\$350$  &amp;    $900,000$ \\
$b$ &amp; $1000~\textrm{km}^2$ &amp; $\$1000$ &amp; $10,000,000$ \\
$c$ &amp; $3500~\textrm{km}^2$ &amp; $\$575$  &amp;    $500,000$ \\
$d$ &amp; $2000~\textrm{km}^2$ &amp; $\$350$  &amp;    $500,000$ \\
\end{tabular}
\end{center}
\begin{enumerate}
\item What is the expected value of $X$?
\item What is the variance of $Y$?
\item Are $X$ and $Y$ independent?  Prove your answer.
\end{enumerate}

\item Let $X$ and $Y$ be random variables on a probability space $(S,P)$.  Show
that if $X$ and $Y$ are independent, then $Var(X + Y) = Var(X) + Var(Y)$.
\emph{Note:} this is an easy proof to find in lots of places.  You may consult
them if you wish, but make sure that your submitted work is your own.

\end{exercises}

\end{document}
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