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%  Homework for the course "Stochastic Calculus", 
%  Fall semester, 2002, Jonathan Goodman.


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\noindent
{\scriptsize Stochastic Calculus, Fall 2002 
(http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)} \hfill 

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\large
Assignment 1.   
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\end{center}

\noindent
Given  September 5, due September 12.\\
Last revised, September 4.
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%           The questions!



\noindent
{\bf Objective:} Review of Basic Probability.
\vspace{.5cm}

\begin{description}

\item[1.]
We have a container with $300$ red balls and $600$ blue balls.  We mix the 
balls well and choose one at random, with each ball being equally likely
to be chosen.  After each choice, we return the chosen ball to the container
and mix again.  
\begin{description}
\item[a.] What is the probability that the first $n$ balls chosen are all blue?
\item[b.] Let $N$ be the number of blue balls chosen before the first
red one.  What is the $P(N=n)$?  What are the mean and variance of $N$.
Explain your answers using the formulae
\begin{eqnarray*}
\sum_{n=0}^{\infty} x^n  & = & \frac{1}{1-x} 
                 \;\;\;\;\;\;\;\;\; \mbox{for $\left|x\right|< 1$}\\
\sum_{n=0}^{\infty} nx^n & = & x \frac{d}{dx} \frac{1}{1-x}  
                             \;\;\;\mbox{for $\left|x\right|< 1$}\\
\mbox{etc.} && 
\end{eqnarray*}
\item[c.] What is the probability that $N=0$ given that $N\leq 2$?
\item[d.] What is the probability that $N$ is an even number?  Count
$0$ as an even number.
\end{description}

\item[2.] A tourist decides between two plays, called ``Good'' (G) and 
``Bad'' (B).  The probability of the tourist choosing Good is $P(G)=10\%$.
A tourist choosing Good likes it (L) with $70\%$ probability ($P(L\mid G)=.7$) 
while a tourist choosing Bad dislikes it with $80\%$ probability
($P(D\mid B) = .8$).  
\begin{description}
\item[a.] Draw a probability decision tree diagram to illustrate
the choices.
\item[b.] Calculate $P(L)$, the probability that the tourist liked the 
play he or she saw.
\item[c.] If the tourist liked the play he or she chose, what is the 
probability that he or she chose Good?
\end{description}

\item[3.] A ``triangular'' random variable, $X$, has probability density
function (PDF) $f(x)$ given by
\begin{displaymath}
f(x) = \left\{ \begin{array}{cl} 2(1-x) & \mbox{if $0\leq x \leq 1$,} \\
                                    0   & \mbox{otherwise.}
               \end{array} \right.
\end{displaymath}
\begin{description}
\item[a.] Calculate the mean and variance of $X$.
\item[b.] Suppose $X_1$ and $X_2$ are independent samples (copies) of $X$ and 
$Y=X_1 + X_2$.  That is to say that $X_1$ and $X_2$ are independent random 
variables and each has the same density $f$.  Find the PDF for $Y$.
\item[c.] Calculate the mean and variance of $Y$ without using the formula
for its PDF.
\item[d.] Find $P(Y>1)$.
\item[e.] Suppose $X_1$, $X_2$, $\ldots$, $X_{100}$ are independent samples
of $X$.  Estimate $Pr(X_1 + \cdots + X_{100} > 34)$ using the central limit
theorem.  You will need access to standard normal probabilities either 
through a table or a calculator or computer program.
\end{description}

\item[4.] Suppose $X$ and $Y$ have a joint PDF 
$$
f(x,y) = \frac{1}{8\pi} \left\{ \begin{array}{cl}
       4 - x^2 - y^2 & \mbox{if $\displaystyle x^2 + y^2 \leq 4$,} \\
        0            & \mbox{otherwise.}
    \end{array} \right.
$$
\begin{description}
\item[a.] Calculate $P(X^2 + Y^2 \leq 1)$.
\item[b.] Calculate the marginal PDF for $X$ alone.
\item[c.] What is the covariance between $X$ and $Y$?
\item[d.] Find an event depending on $X$ alone whose probability depends
on $Y$.  Use this to show that $X$ is not independent of $Y$.
\item[e.] Write the joint PDF for $U=X^2$ and $V=Y^2$.
\item[f.] Calculate the covariance between $X^2$ and $Y^2$.  It may be
easier to do this without using part e.  Use this to show, again,
that $X$ and $Y$ are not independent.

\end{description}



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