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\title{Method of Moments}
\author{Worksheet 15\footnote{\copyright 2019 Joseph C Watkins}}
\date{}
\maketitle

\begin{enumerate}
\item Assume that $X$, the value an unfair die,  has the following probability mass function.
\begin{center}
\begin{tabular}{c|cccccc}
$x$&1&2&3&4&5&6 \\ \hline
$f_X(x|\alpha)$&$\frac{1}{6}(1-3\alpha)$&$\frac{1}{6}(1-2\alpha)$&$\frac{1}{6}(1-\alpha)$&$\frac{1}{6}(1+\alpha)$&$\frac{1}{6}(1+2\alpha)$&$\frac{1}{6}(1+3\alpha)$ \\
\end{tabular}
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\begin{enumerate}
\item What values for $\alpha$ are possible?
\item Find the mean $\mu=E_\alpha X$ and the variance $\sigma^2 = \hbox{Var}_\alpha(X)$.
\item Let $\bar X$ be the average of 100 dice rolls find the mean $E_\alpha \bar X$ and the variance $ \hbox{Var}_\alpha(\bar X)$.
\item For independent observations, $\mathbf{x} =(x_1,x_2,\ldots, x_{100})$, of this random variable from a given value of $\alpha$, give $\hat\alpha$, the method of moments estimate.
\item Find $ E_\alpha \hat\alpha(X_1,\cdots X_{100})$. Is this estimator biased?
\item Provide 10,000 simulations for the case $\alpha = -1/10$. Find the mean and standard deviation of these simulations. How does this compare to your results in parts (e)?
\item Make a histogram of the values for $\hat\alpha$ and provide a desciption of the histogram.
\end{enumerate}

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