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% Extra Credit 6501 ******************************************************
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% Submitted:      Dr. Eric Suess, Statistics 6501, Winter 2012
% Submitted Date: February 5th, 2012
% Accepted Date:  February 5th, 2012
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\markright{Statistics 6401, Fall 2012}

\large
\begin{center}
\textbf{Extra Credit}
\end{center}

Direct proof that $\bar{X}$ and $S^2$ are independent when sampling from the $N(\mu,\sigma^2)$ distribution.

Let $X_1, X_2$ independent $N(\mu,\sigma^2)$ random variables.  (A random sample of size $n=2$.)

\begin{enumerate}
\item
Show that $Y_1 = X_1 + X_2$ and $Y_2 = X_2 - X_1$ are independent.
\item
What is the distribution of $Y_1$?  What is the distribution of $Y_2$?
\item
Show that $W_1 = \frac{1}{2}Y_1$ and $W_2 = \frac{1}{2}Y_2$ are independent.

Note:
\begin{eqnarray*}
W_1 &=& \frac{1}{2} Y_1 \\
&=& \frac{1}{2}(X_1 + X_2) \\
&=& \bar{X}
\end{eqnarray*}
and
\begin{eqnarray*}
W_2 &=& \frac{1}{2} Y_2 \\
&=& \frac{1}{2}(X_2 - X_1) \\
&=& \frac{1}{2} X_2 - \frac{1}{2} X_1 \\
&=& X_2 - \frac{1}{2}X_1 - \frac{1}{2} X_2 \\
&=& X_2 - \left( \frac{X_1 + X_2}{2} \right) \\
&=& X_2 - \bar{X}
\end{eqnarray*}
(So $\bar{X}$ and the $n-1$ deviations from the sample mean are independent.)
\item
What is the distribution of $W_1$?  What is the distribution of $W_2$?
\item
Show that since $W_1$ and $W_2$ are indenpendent that $W_3 = X_1 - \bar{X}$ is also
independent of $W_1$.

Note:
\begin{eqnarray*}
X_1 - \bar{X} + X_2 - \bar{X} &=& 0
\end{eqnarray*}
(So $\bar{X}$ and the first deviation $X_1 - \bar{X}$ are also independent.)
\item
Argue that $\bar{X}$ and $S^2 = \frac{1}{n-1} \sum_{i=1}^2 (X_i - \bar{X})^2$ are independent
for a random sample of size $n = 2$ from the $N(\mu,\sigma^2)$ distribution.
\item
What is the distribution of $\bar{X}$? What is the distribution of $S^2$?
\item
Develop the same results for $n = 3$.
\item
Develop the same result for a sample of size $n$.
\end{enumerate}

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