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\renewcommand{\sheweek}{MT2006/2007 Exercise 4}
\begin{center}
\Huge \textbf{Exercise 4}
\end{center}
\vspace{1cm}

\begin{enumerate}
\item The probability that an insect lays $r$ eggs is Poisson with mean
$\lambda $, and the probability that an egg develops is $p$.
Assuming independence between the eggs, find the probability that
$k$ eggs survive, $k = 0,1,2,\dots$.   (We have a joint
distribution for $(K, R)$ and want the marginal probability
function for $K$.)

\item Suppose that the number of insurance claims that a particular policyholder
makes in one year has a Poisson distribution with mean $\Lambda $,
and that over the large population of policyholders $\Lambda $ has
a Gamma distribution. Find the probability of $x$ claims in one
year $x = 0$, 1,...   by a policyholder chosen at random from the
population of policyholders.   (There is a joint distribution for
$(X, \Lambda )$, and we require the marginal distribution of $X$.)
\item Suppose that a bus is $X$ minutes late, where $X$ has an exponential
distribution with mean $1/\Theta $, where $\Theta $ varies
randomly with the density of the traffic according to the Gamma
density function

$$
f(\theta ) = \lambda ^\alpha \theta ^{\alpha -1}e^{-\lambda \theta
}/\Gamma(\alpha ).
$$

What is the density function of $X$ over all values of $\Theta$?
(Write down the joint density function for $(X, \Theta )$ and
integrate out over values of $\Theta $ to get the marginal density
function of $X$.)

\item  If $X$ and $Y$ have joint distribution given by
$$ f_{X,Y}(x,y) = \left\lbrace \begin{array}{ll}
                       2 & 0 < x < y < 1 \\
                       0 & \mbox{elsewhere}
                      \end{array}\right.
$$

find the covariance of $X$ and $Y$, and the conditional density
function of $Y$ given $X = x$.
\item
 Let the joint density function for $X, Y$ be

$$
f_{X,Y}(x,y) = \left\lbrace \begin{array}{ll}
                             8xy& 0 < y < x < 1\\
                             0&\mbox{elsewhere}
                             \end{array}\right.
$$

\begin{enumerate}
\item Find $E[Y|X=x].$
\item Find $\var[Y|X=x].$
\item Find $E[XY|X=x]$, and hence the covariance of $X$ and $Y.$
\end{enumerate}

\item Suppose that the joint density function of $(X,Y)$ is

$$
f_{X,Y}(x,y) = \left\lbrace\begin{array}{ll}
                            1 - a(1-2x)(1-2y)& 0<x<1, 0<y<1\\
                            0& \mbox{elsewhere}
                            \end{array}\right.
$$
where $-1 < a < 1$.

Prove that $X$ and $Y$ are independent if and only if $X$ and $Y$
are uncorrelated.

\item Find the density function of the sum of two independent
exponential random variables with the same mean. Do the same for
the case where the means are different.

\item Let
\[
Y = \left\{\begin{array}{ll}
            \sum_{i=1}^N X_i & \mbox{ if } N\ge 1\\
            0 & \mbox{ if } N=0
           \end{array} \right.
\]
where $\{X_i\}$, $i=1, 2,\dots$ is a sequence of iid random
variables and $N$ is a discrete random variable taking values $0,
1,\dots$ independent of $\{X_i\}$.  Let $K_Y(t), K_N(t), K_X(t)$
denote the cumulant generating functions of $Y$, $N$ and $X_i$
respectively. Prove that
\[
K_Y(t) = K_N[K_X(t)].
\]
Furthermore, suppose that $P(N = i) = p^{i-1}(1-p)$ for $i = 1,
2,\dots$ and the density of $X_i$ is given by $\alpha^2 x
e^{-\alpha x}$ for $x>0$. Prove that $Y$ has density function
\[
\frac{\alpha (1-p)}{2\sqrt{p}}\left( e^{-\alpha (1-\sqrt{p})y} -
e^{-\alpha (1+\sqrt{p})y}\right),
\]
for $y > 0$.

\item Suppose that $X$ and $Y$ are bivariate normally distributed
with zero means, unit
variances and correlation coefficient $\rho $.   Show that the
correlation coefficient between $X^2$ and $Y^2$ is $\rho ^2$.
(Use bivariate cumulants.)

\item Find the joint moment generating function of $X$ and $X^2$
where $X$ is a
standard normal random variable.

\item Find the moment generating function for the product of two
independent standard normal random variables, $X$ and $Y$, by
first finding the conditional moment generating function of $XY$
for a fixed value of $X$. Hence or otherwise show that if $U$,
$V$, $X$ and $Y$ are independent standard normal random variables,
then $UV+XY$ has a Laplace distribution.

\item If $X$ and $Y$ are iid exponential random variables, find
the joint density of $U=X/Y$ and $V=X+Y$.

\item Suppose that $(R, \Theta)$ are the polar coordinates of the
random variables $(X,Y)$ and that $R$ has the density function
$f_R(r)$ and is independent of $\Theta$ which has a uniform
distribution on $(0,2 \pi)$. Find an expression for the joint
density of $(X,Y)$ in terms of $f_R$. Interpret your result
geometrically.

\item If $(X$, $Y)$ have some joint distribution, then one might try to define the
concept of `$Y$ stochastically bigger than $X$' by the property
$F_X(z) > F_Y(z)$ for all real $z$.   Show that
\begin{enumerate}
\item If $F_X(z) > F_Y(z)$ for all $z$, then $E[Y] > E[X]$.   You may assume $X$
and $Y$ are continuous random variables.
\item If $F_X(z) > F_Y(z)$ for all $z$, then $P[Y > X] > 0$.
\item If $X$ and $Y$ are independent continuous random variables, and $F_X(z) > F_Y(z)$ for all $z$, then $P[Y>X]>0.5$.
\end{enumerate}


\end{enumerate}

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