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\pset{10}{May 4, 1995}{In recitation, May 12}

\bigskip

\begin{problems}

% Problem 1

\problem 
\bparts
\ppart Let $X$ and $Y$ be two random variables defined on the same probability space.  Show that:
\[
E(X) = \sum_k {\rm Pr}(Y=k)E(X \mid Y=k)
\]
\ppart Consider the following random experiment.  We have $n$ (fair) dice in
a bin.  We shake the bin and tip it over.  Some of the dice roll out (it is
equally likely that any number from $1$ to $n$ will roll out).  We then sum
up all of the values on the top faces of the dice that roll out. What is the
expected value of this sum?  
\eparts

% Problem 2

\problem
Suppose we toss a fair coin $n$ times.  We define the random variables $X_i$
($i=1, \ldots, n$) by:
\[
X_i = \left\{ \begin{array}{cl} 1 & \mbox{if the $i$th toss is heads} \\ 0 &
\mbox{otherwise} \end{array} \right.
\]
Let $X = \sum_{i=1}^n X_i$.  Compute the following (where $a,b \in \{0,1\}$
and $0 \leq k \leq n$):
\bparts
\ppart ${\rm Pr}(X_i = a \mbox{ \& } X_j = b \mid X = k)$
\ppart ${\rm Pr}(X=k \mid X_i = a \mbox{ \& } X_j = b)$
(where $a,b \in \{0,1\}$ and $0 \leq k \leq n$)
\eparts

% Problem 3

\problem  

You are given a wheel (like a roulette wheel) which contains the numbers
from $1$ to $2n$ arranged in some way. The wheel has a spinner and on each
spin you get the two numbers at the opposite ends of the spinner.  How would
you arrange the numbers on the wheel to maximize the expected value of:

\bparts
\ppart The sum of the numbers chosen?
\ppart The product of the numbers chosen?
\eparts

% Problem 4
\problem 
Let $S$ be a sample space.  Say random variable $X: S \rightarrow \naturals$ has 
distribution
\[
X = \left\{ \begin{array}{cl} 2 & \mbox{with probability $1/4$,} \\
                             3 & \mbox{with probability $1/4$,} \\
			     5 & \mbox{with probability $1/2$;} \end{array}
\right.
\]
and random variable $Y:S\rightarrow \naturals$ has distribution
\[
Y = \left\{ \begin{array}{cl} 1 & \mbox{with probability $2/3$,} \\
                             4 & \mbox{with probability $1/3$;} \end{array}
\right.
\]
where $X$ and $Y$ are independent.

Let random variable $Z:S\rightarrow \naturals$ be given by $Z = \max(X,Y)$.
\bparts
\ppart What is the expected value of $Z$?
\ppart What is the variance of $Z$?
\eparts

% Problem 5

\problem
We are given a random vector of $n$ distinct numbers.  We then
determine the maximum of these numbers by the following procedure:

Pick the first number.  Call it the ``current maximum''.  Go through
the rest of the vector (in order) and each time we find a number $x$
that exceeds our ``current maximum'', we update the ``current maximum''
with $x$.

What is the expected number of times we update the ``current maximum''?

% Problem 6

\problem

\bparts
\ppart Show that if random variables $X$ and $Y$ are independent then ${\rm
Cov}(X,Y)=0$

\ppart Show that having ${\rm Cov}(X,Y)=0$ does not necessarily mean that
$X$ and $Y$ are independent.

\ppart Let $X_1,\ldots,X_n$ be random variables.  Prove that
\[
{\rm Var}(X_1+\ldots+X_n)=\sum_{i=1}^n {\rm Var}(X_i) + 2\sum_{i<j} {\rm Cov}(X_i,X_j)
\]
\eparts

\end{problems}

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