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\date{March 10, 2005}

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\title{Stat 324: Lecture 13\\
Sampling distributions}

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\author{Moo K. Chung\\
mchung@stat.wisc.edu}

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\section*{Concepts}
\begin{enumerate}

\item Random variables $X_1,\cdots, X_n$ form a {\em random sample}
if they are independent and identically distributed (i.i.d.). A {\em
statistic} is a function of the random sample, i.e.
$h(X_1,\cdots,X_n)$. The distribution of the statistic is called the
{\em sampling distribution}.

\item Suppose $\mathbb{E} X_i = \mu, \mathbb{V} X_i = \sigma^2$. Then
$\mathbb{E} \bar X = \mu, \mathbb{V} \bar X = \sigma^2/n$. This can
be shown from the fact that $\mathbb{E} (\sum_{i=1}^n X_i) =
\sum_{i=1}^n \mathbb{E} X_i, \mathbb{V} (\sum_{i=1}^n X_i) =
\sum_{i=1}^n \mathbb{V} X_i$.

\item If a random sample $X_i \sim N(\mu,\sigma^2)$, then $\bar X \sim N(\mu,
\sigma^2/n)$.

\item {\em Central limit theorem.} Consider a random sample $X_1,\cdots,X_n$
with mean $\mu$ and variance $\sigma^2$. For sufficiently large $n$,
$\bar X \sim N(\mu,\sigma^2/n)$. The larger the value of $n$, the
better the approximation. Similarly $Z=\frac{\bar X -
\mu}{\sigma/\sqrt{n}} \sim N(0,1).$

\end{enumerate}
\section*{Problems}
\begin{enumerate}
\item Let $X_1,\dots,X_n \stackrel{i.i.d.}{\sim} N(\mu,\sigma^2)$. Then are $\bar X$ and $X_j - \bar X$ independent?

{\it solution.} $\bar X \sim N(\mu,\frac{\sigma^2}{n})$. Since $X_j
- \bar X$ is a linear combination of $X_i$'s, it is also distributed
as normal. Any uncorrelated normal distributions are independent. So
we only need to check if ${\bf Cov}( \bar X,X_j-\bar X)=0$. \bq {\bf
Cov}(\bar X, X_j - \bar X) &=& {\bf Cov}(\frac{1}{n}\sum_{i=1}^n
X_i, X_j -
\frac{1}{n}\sum_{k=1}^nX_k) \\
&=& \frac{1}{n}\sum_{i=1}^n {\bf Cov}(X_i,X_j)
-\frac{1}{n^2}\sum_{i,k=1}^n {\bf Cov}(X_i,X_k) = \frac{1}{n}\mathbb{V}(X_j) - \frac{1}{n^2}\sum_{i=1}^n \mathbb{V}(X_i)\\
&=&  \frac{1}{n}\mathbb{V}(X_j) - \frac{1}{n}\mathbb{V}(X_j)=0\eq


\item The service times for customers coming through a checkout counter in a retail store are independent random variables with mean 2 minutes and variance 5.
(Hint: Apply the central limit theorem to answer the following questions)\\
(a) Approximate the probability that 100 customers can be served in less than 3 hours of total service time.\\
(b) Find the maximum number of customers that can be served in less than 3 hours with probability 0.9.\\
(c) Approximate the probability that 100 customers can be served in
between 3 hours and 3 hours 40 minutes.

\end{enumerate}
{\bf Assigned Problems:} Exercise 5.48. Exercise 5.50. Exercise
5.90. {\bf Homework 4} due on March 17th. 9:30AM. Solve assigned
problems in Lecture 11-14. Please read Chapter 6.
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