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\handout{quiz3}{Quiz 3}{May 7, 1996}

This is an OPEN BOOK Quiz: you may refer to books, handouts, or your own
course notes.  (Calculators are not allowed.)  You may use in your problem
solutions any of the mathematical results given in lectures, recitations,
or the course text.  
There are six problems of roughly equal weight to
complete within one hour and 20 minutes.  (Finish by 3:55 PM.)
Good luck!

\begin{problems}

\problem
Let $X$ be a $\{1,0,-1\}$ valued random variable and $Y$ a $\{1,-1\}$
valued random variable with joint distribution given by the following
table:

\[
\begin{array}{|r|r|c|} \hline
j     & k        & P((X=j) \wedge (Y=k))\\ \hline\hline
1 & 1 & a \\ \hline
0 & 1 & b \\ \hline
-1 & 1 & c \\ \hline
1 & -1 & d \\ \hline
0 & -1 & e \\ \hline
-1 & -1 & f \\ \hline
\end{array}
\]
For example, $P((X=0) \wedge (Y=1))= b$.

For each of the quantities in the box on the left below, indicate the
number of the expression for it from the box on the right.  For example,
the first entry is filled in with the answer ``(14)''. (Write your answers
in your blue answer book.)\\
$\begin{array}{|c|c|c|}\hline

0. & P((X=0) \wedge (Y=1)) & \mbox{(14)} \\ \hline
1. & P(X=0) & \\ \hline
% P(Y=-1) & \\ \hline
% 2. & P((Y=1) \mid (X=1)) & \\ \hline
2. & P((Y=1) \mid (X\geq 0)) & \\ \hline
% E(X) & \\ \hline
3. & E(Y) & \\ \hline
% 5. & E(X^{2}) & \\ \hline
4. & \mbox{Var}(X) & \\ \hline
5. & E(X\cdot Y) & \\ \hline
6. & \mbox{Cov}(X,Y) & \\ \hline
\end{array}$\qquad{\begin{tabular}{|l|}\hline
(1) $a$\\
(2) $a+b$\\
(3) $b+e$\\
(4) $d+e+f$\\
(5) $a+b+d+e$\\
(6) $a+d-c-f$\\
(7) $a+c+d+f$\\
%or 1-b-e
(8) $a+f-c-d$\\
(9) $2(a+b+c)-1$\\
% or a+b+c-d-e-f or 1-2(d+e+f)
(10) $a/(a+d)$\\
(11) $(a+b)/(a+b+d+e)$\\
% or (a+b)/(1-c-f)
(12) $a+c+d+f-(a+d-c-f)(1-b-2c-e-2f)$\\
% or 1-b-e-(a+d-c-f)^{2}
(13) $a+f-c-d-(a+d-c-f)(a+b+c-d-e-f)$\\
% or a+f-c-d-(a+d-c-f)(1-2(d+e+f))
(14) $b$\\ \hline
\end{tabular}}

\problem 
A round pizza is divided into four equal slices (but still left on the tray).
Then nine pepperoni slices are placed on the pizza.
Each pizza slice gets an integral number, possibly zero, of pepperoni slices.

\bparts
\ppart
How many different ways are there to arrange the nine pepperoni slices
on the four (distinguishable) pizza slices?
% \ppart 
% Argue that there is no way to arrange the nine pepperoni slices on
% the pizza so that it has any sort of rotational symmetry. (That is,
% argue that rotation by any multiple of 90 degrees must give a
% different arrangement.)  What numbers, other than nine, have this
% property?
\ppart
How many different ways are there to arrange the nine pepperoni slices
on the four pizza slices, if two arrangements are considered ``the
same'' if one can be obtained from the other by a rotation by 
some multiple of 90 degrees?
(You may take it as a fact that a rotation of the pizza by 90, 180, or 270
degrees always gives a different-looking distribution of the nine pepperoni
slices among the four pizza slices.)
\eparts

\problem
Mad Cow disease affects only 1 cow in a million.  A new test for the
disease always reports correctly when a cow is infected, but has a
probability $q$ of mistakenly reporting that an uninfected cow is infected.

Let $P(M|T)$ denote the probability that a randomly chosen cow has the disease
(event $M$) given that the test says the cow is infected (event $T$).  

\bparts
\ppart Write an expression for $P(M|T)$.  
\ppart How small does $q$ have to be for $P(M|T) \geq\frac{1}{2}$?
\eparts

\problem
A pollster inteviews $n$ people, who answer independently with
probability $p$ in favor of Clinton over Dole.  The pollster estimates
$p$ as the fraction of people who respond in favor of Clinton.  His
boss tells him to interview enough additional people to lower the
standard deviation $\sigma$ of his estimate of $p$ to $\sigma/5$.
\bparts
\ppart How many {\em additional} people should he interview?
\ppart What does Chebyshev's Theorem imply about the probability that the 
second estimate of $p$ will be in error by more than $\sigma$.
(Explain briefly.)
\eparts

% Too easy -- RLR
% \problem
% 
% Let $Y$ be any random variable and $h$ any non-negative
% real function.
% Show that for all $t>0$,
% \[\P(h(Y) \geq t) \leq {E[h(Y)] \over t} \]

\problem
Let $X$ and $Y$ be arbitrary random variables. Prove the following:
\bparts
\ppart $Cov(c\cdot X,d\cdot Y)=cd\cdot Cov(X,Y)$ for any constants $c$,$d$.
\ppart $\mbox{Cov}(X,Y+Z) = \mbox{Cov}(X,Y) + \mbox{Cov}(X,Z)$.
% \ppart $\mbox{Cov}(X+Y,X-Y) = \mbox{Var}(X) - \mbox{Var}(Y)$.
%
% \ppart $Cov(c,X) = 0$ for any constant $c$.
% \ppart $Var(X+c)=Var(X)$ for any constant $c$.
% \ppart If $X = X_1 + X_2 + \cdots + X_n$, then
% \[
%        Var(X) = \sum_{1\le i \le n \atop 1\le j \le n} Cov(X_i,X_j)\ .
% \]
% \ppart  $ Var(X+E(Y))+Var(E(X)-Y) = (Var(X+Y)+Var(X-Y))/2 $
\eparts

\problem
Let $S$ be a discrete sample space, and let $X:S\to\nats$ and
$Y:S\to\nats$ be two random variables.
(Here $\nat$ denotes the set of natural numbers $\{0,1,2,\ldots\}$.)

% \bparts
% \ppart
Let $A$ and $B$ be two subsets of natural numbers. Write $(X\in A)$
for the event that $X$ assumes a value in $A$, and likewise $(Y\in
B)$.

Prove that if $X$ and $Y$ are independent random variables, then
$(X\in A)$ and $(Y\in B)$ are independent events.

% \ppart
% If $f, g:\nats\to\nats$ are functions, then $f(X)$ and $g(Y)$ are also
% random variables on $S$.
% 
% Prove that if $X$ and $Y$ are independent, then $f(X)$ and
% $g(Y)$ are independent as well. ({\em Hint:} use part (a).)
% 
% \eparts

\end{problems}

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