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\title{The Expected Value\footnote{\copyright 2019 Joseph C Watkins}}
\author{Worksheet 10}
\date{}
\maketitle

\begin{enumerate}
\item Let $X$ and $Y$ be discrete random variables with mass functions
\begin{center}
\begin{tabular}{c|cccccc}
$x$&1&2&3&4&5&6 \\ \hline
$f_X(x)$&0&0.1&0.2&0.3&0.3&0.1\\
\end{tabular}\qquad
\begin{tabular}{c|cccccc}
$y$&1&2&3&4&5&6 \\ \hline
$f_Y(y)$&0.2&0.2&0.2&0.2&0.2&0
\end{tabular}
\end{center}

\begin{enumerate}
\item Find $P\{X\le 3\}$.
\item Find $P\{Y>2\}$.
\item Find $EX$ and $EY$.
\item On a common graph, draw the survival function for $X$ and $Y$.
\item Find the area between the survival functions and check that it is equal to $E[X-Y]$.
\end{enumerate}

\item The {\bf Pareto random variable} with parameters $\alpha>0$ and $\beta>0$ has probability density function
$$f_X(x) = \frac{\beta\alpha^\beta}{x^{\beta+1}}, \quad \alpha < x< \infty.$$
\begin{enumerate}
\item Verify that $f_X$ is a density function.
\item Find $P\{X > 2\alpha\}.$
\item Find the mean and variance of $X$. What restriction do you have on $\beta$ in computing the mean and variance (a different restriction for each)?
\item Use the probability transform to simulate 1000 Pareto random variables with $\alpha=1$ and $\beta = 4$ and find their sample mean and variance. Compare this to the values in part (c).
\end{enumerate}
\end{enumerate}

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