\renewcommand{\coddate}{\small \today}
\renewcommand{\sheweek}{MT2006/2007 Exercise 5}
\begin{center}
\Huge \textbf{Exercise 5}
\end{center}
\vspace{1cm}

\begin{enumerate}

\item
(i) Let $b_i = \alpha a_i + \beta$, $i =1 , \ldots , n$,
and $\bar a = {1\over n}\sum_{i=1}^n  a_i$,
$\bar b ={1\over n} \sum_{i=1}^n        b_i$.
Show that $\bar b = \alpha \bar a + \beta$, and
\[
\sum_{i=1}^n (b_i - \bar b)^2
= \alpha^2 \sum_{i=1}^n (a_i - \bar a)^2.
\]
(Note that the sample variance of $\{ b_i \}$ is independent
of $\beta$.)

(ii) Let $\bar a = {1\over n} \sum_{i=1}^n  a_i$ and
$\bar b =  {1\over n}\sum_{i=1}^n  b_i$.
Show that

\[
\sum_{i=1}^n (a_i - c) (b_i - d)
= \sum_{i=1}^n (a_i - \bar a)(b_i - \bar b)
+ n (\bar a -c)(\bar b -d)
\]
for any constants $c$ and $d$, and
\[
\sum_{i=1}^n (a_i - \bar a)(b_i - \bar b)
= \sum_{i=1}^n  a_i b_i  - n \bar a \bar b
\]

\item
Let $\{ X_1, \ldots, X_n\}$ and $\{ Y_1, \ldots , Y_m\}$ be two independent
samples, where $X_1, \ldots, X_n$ are independent observations from
$N(\mu_1, \sigma^2 )$, and $Y_1, \ldots , Y_m$ are independent observations
from $N(\mu_2, \sigma^2 )$. Define
\[
\bar X = {1 \over n} \sum_{i=1}^n X_i, \;\;\;\;
s_x^2 = {1 \over n-1} \sum_{i=1}^n (X_i - \bar X)^2,
\]
\[
\bar Y = {1 \over m} \sum_{i=1}^m Y_i, \;\;\;\;
s_y^2 = {1 \over m-1} \sum_{i=1}^m (Y_i - \bar Y)^2.
\]
Find the sampling distributions of (i) $\bar X - \bar Y$,
(ii) $T \equiv (n-1)s_x^2 + (m-1)s_y^2$, and (iii)
${\bar X - \bar Y \over \sqrt{T} }
\sqrt{ nm(n+m-2) \over m+n}$ when
$\mu_1 = \mu_2$.

\item
A finite population has $N_a$ individuals of type A and $N_b$ individuals
of type $B$. The overall number $N_a+N_b$ is equal to 100, but $N_a$ is unknown.
If a sample of 5 individuals is taken without replacement, write down the likelihood
function
for $N_a$ when the observed sequence is
(i) $A, B, A, B, A$, or (ii) $A, A, B, B, A$.
Compare them with the likelihood function when the precise sequence of the outcomes
is not available, but it is known that three elements are $A$ and two others are $B$,
and comment on your findings.


\item Given a random sample $Y_1,...,Y_n$ from the following
distributions, find a simple sufficient statistic using the
factorisation criterion.
\begin{enumerate}
\item Poisson with mean $\lambda$
\item Normal with mean 0 and variance $\sigma ^2$
\item Geometric
\item Uniform on $(\theta _1, \theta _2)$
({\small Hint:
$f_Y(y; \theta_1, \theta_2) = {1 \over \theta_2 -\theta_1} I_{(\theta_1, \infty)}(y)
I_{(-\infty, \theta_2)}(y).$
})
\item density $f_Y(y;\theta) = \exp(-(y-\theta))$ for $y > \theta$ (Use indicator functions, and take care with the factorisation)
\end{enumerate}

\end{enumerate}
