\renewcommand{\coddate}{\small \today} %Put in date you modify the file
\renewcommand{\sheweek}{MT2006/2007 Exercise 3}
\begin{center}
\Huge \textbf{Exercise 3}
\end{center}
\vspace{1cm}

\begin{enumerate}
\item Find the moment generating function of $N(\mu, \sigma^2)$. Find
the mean and the variance of $Y = e^X$ if $X \sim N(\mu,
\sigma^2)$. ($Y$\ has the Lognormal distribution, very popular as
a skew distribution for positive variables.)

\item State and prove the Markov Inequality.  Use it to establish that,
if $Y$ is a random variable with MGF $M_Y(t)$, then for $\gamma
>0$ and $t>0$,
\[
P(Y \ge \gamma) \le \frac{M_Y(t)}{e^{\gamma t}}.
\]
Hence show that if $Z$ has a standard normal distribution, then
for $\gamma \ge 0$,
\[
P(Z \ge \gamma) \le e^{-\gamma^2/2}.
\]

\item \begin{enumerate}
\item Let $X$ be a random variable with MGF $M_X(t)$. Find an expression for the cumulant
generating function $K_X(t)$ as a function of the moments $m_r$.
Hence find expressions for $\kappa_2$ and $\kappa_3$ in terms of
these moments.
\item Let $Y = X - m_1$. Find expressions for $\kappa_2$,
$\kappa_3$, $\kappa_4$ and $\kappa_5$ as functions of the moments
of $Y$.
\item Now let $Z = Y/\sigma$. Find an expression for $K_Z(t)$ as a
function of $K_Y(t)$. Hence find $\kappa_2$, $\kappa_3$ and
$\kappa_4$ for $Z$ as functions of the moments of $Y$.
\item Finally, suppose that $X$ is distributed $N(\mu,\sigma^2)$. Write down
$K_X(t)$ in this case. Use your results in (a) to find $m_2$ and
$m_3$ for $X$.
\end{enumerate}

\item Find the moment generating function of the Double Exponential or Laplace
distribution with density function

$$
f_X(x) = \frac{1}{2}e^{-\mid x\mid }\ \ \  -\infty  < x < \infty ,
$$

and hence its first four cumulants.   (It has been suggested that the
Laplace distribution is more typical of naturally occurring physical
measurements than the normal distribution.)


\item  If $H(x)$ and $G(y)$ are distribution functions, for which of the following
definitions is $F(x,y)$ a joint distribution function?
(Sometimes one can give a simple description of the joint
distribution of $X$ and $Y$ which leads to the form given.   The
notations $\max (a,b)$ and $\min (a,b)$ mean the largest of $a$,
$b$ and the smallest of $a$, $b$ respectively.)

\begin{enumerate}
\item $F(x,y) = H(x) + G(y)$
\item $F(x,y) = H(x)G(y)$  (use independence)
\item $F(x,y) = \max [H(x),G(y)]$
\item $F(x,y) = \min [H(x),H(y)]$ (consider $X = Y$).
\end{enumerate}
\end{enumerate}
