% BTS exercise 1

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%DOCUMENT
\begin{document}

\begin{center}
\huge ST402 Principals and Methods of Statistical Practice \\
\huge Mid-term Test \\
\end{center}

\Large Answer as much of this as you can in the 45 minutes
available. You may consult your notes but not your colleagues. All
questions will be marked out of 10 (although question 4 may be
slightly longer than the others).

\begin{enumerate}
%
% Qu 1
%
\item I have an urn with 3 disks of which 2 are green and 1 is
red. I toss a fair coin twice and count the number of heads; $X$
is the number of heads. I then draw $X$ disks from the urn without
replacement; $Y$ is the number of red disks that I draw.
%
   \begin{enumerate}
   \item Find $E(X)$ and $E(Y|X)$. Hence find $E(Y)$.
   \item Find the probability mass function of $Y$.
    \end{enumerate}
%
%
% Qu 2
%
\vspace{1cm}
\item Let $X$ and $Y$ be random variables with joint density function
\begin{equation} \label{joint:eq}
f_{X,Y}(x,y) = \left\{ \begin{array}{ll} kx, & 0 < x < y < 1, \\
0, & {\rm otherwise.} \end{array}\right.
\end{equation}
\begin{enumerate}
\item Find $k$.
\item Write down the density of $X|Y=y$ and hence find $\var(X|Y=
\frac12)$.
\end{enumerate}
%
\vspace{1cm}
% Qu 3
%
\item Let $Y$ and $Z$ be independent standard normal random
variables.
\begin{enumerate}
\item Show that the moment generating function of $Y$ is $M_Y(t) =
e^{t^2/2}$ and hence write down the joint MGF $M_{Y,Z}(t,u)$.
\item Let $W=Y$ and $X= \rho Y + \sqrt{(1-\rho^2)} Z$. Derive the
joint moment generating function of $W$ and $X$. Show that $X$ is
a standard normal random variable.
\end{enumerate}
%
%
\newpage
% Qu 4
%

\item Suppose that $X_j \sim i N(0, \sigma^2_X)$  and
$Y_j \sim i N(\mu_Y, \sigma_Y^2)$ for $j=1,\dots,n$.
\begin{enumerate}
\item Find the $r^{\rm th}$ moment of $X$.
\item Find the expected value and variance of the estimator of
$\sigma_X^2$
$$
\hat{\sigma}_X^2 = \frac1n \sum_{j=1}^n X_j^2
$$
\item Find the expected value of the the estimator of $\sigma_Y^2$
$$
\hat{\sigma}_Y^2 = \frac1n \sum_{j=1}^n (Y_j - \overline{Y})^2
$$
where $\overline{Y} = \frac1n \sum_{j=1}^m Y_j$.
\end{enumerate}
\end{enumerate}
\end{document}
