% BTS exercise 1

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\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 roll a fair die (X is the number shown) and then
spin a fair coin the number of times indicated by the die and
count the number of heads (Z is the number of heads).
%
   \begin{enumerate}
   \item  What well known distribution does $Z|X=x$ have? Specify
   the parameters of this distribution. Use iterated expectation
   to find $E(Z)$.
   \item  Write down an expression for $P(Z=5|X=x)$. Use the law
   of total probability to find $P(Z=5)$.
    \end{enumerate}
%
%
% Qu 2
%
\item Consider the function
\begin{equation} \label{joint:eq}
f_{X,Y}(x,y) = \left\{ \begin{array}{ll} 3y, & 0 < x < y < 1, \\
0, & {\rm otherwise.} \end{array}\right.
\end{equation}
\begin{enumerate}
\item Show that this is a density function.
\item If $X$ and $Y$ are random variable with joint density given
by \eqref{joint:eq}, find $E(Y|X=x)$.
\end{enumerate}
%
% Qu 3
%
\item Consider the transformation
\begin{eqnarray}
x(u,v) & = & u, \label{x:eq}\\
y(u,v) & = & \rho u + \sqrt{(1-\rho^2)} v, \label{y:eq}
\end{eqnarray}
where $|\rho| < 1$.
\begin{enumerate}
\item Show that the Jacobian of the inverse transformation is
$1/\sqrt{(1-\rho^2)}$.
\item Let $U$ and $V$ be independent standard normal random
variables. Write down the joint density of $U$ and $V$. Define
transformed variables using equations \eqref{x:eq} and
\eqref{y:eq}; $X = x(U,V)$ and $Y = y(U,V)$. Using the change of
variable formula, derive the joint density of $X$ and $Y$.
\end{enumerate}
%
%
% Qu 4
%
\item Suppose that $X_1,\ldots,X_n$ is a sequence of independent
identically distributed random variables with $E(X) = \mu <
\infty$. Let $S_n = \sum_{j=1}^n X_j$ and $Y_n = \frac{1}{n} S_n$.
\begin{enumerate}
\item Show that the moment generating function of $Y_n$ is given
by
$$
M_{Y_n}(t) = M_{S_n}(t/n)
$$
and hence show that the cumulant generating function is
$$
K_{Y_n}(t) = n K_X(t/n)
$$
where $M_{S_n}$ is the moment generating function of $S_n$ and
$K_X$ is the cumulant generating function of $X$. [Hint: you will
need an expression for $M_{S_n}$ in terms of $M_X$.]
\item Suppose that $W$ is a degenerate random variable with all of
its mass concentrated at $\mu$. Thus, the probability mass
function of $W$ is given by
$$
f_{W}(w) = \left\{\begin{array}{ll} 1, & w = \mu, \\
0, & {\rm otherwise.} \end{array}\right.
$$
Show that the cumulant generating function of $W$ is $K_W(t) = \mu
t$.
\item Show that $K_{Y_n}(t) \longrightarrow \mu t$ as $n
\longrightarrow \infty$ [recall $\mu = E(X)$]. Draw the
appropriate conclusions about convergence of $Y_n$.
%
\end{enumerate}

\end{enumerate}
\end{document}
