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\begin{document}
\bigskip
Probability Theory 407\hfill Steven Heilman\\
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Please provide complete and well-written solutions to the following exercises.

Due February 2, 2PM PST, to be uploaded in blackboard as a single PDF document (in the Assignments tab).

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\begin{center}
{\Large Homework 3}
\end{center}

\vspace{.5cm}


\begin{exercise}
Let $\Omega=[0,1]\times[0,1]$ so that $\Omega\subset\R^{2}$.  Define a probability law $\P$ so that, for any set $A\subset\Omega$, $\P(A)$ is defined to be the area of $A$.  Let $0\leq a_{1}\leq a_{2}\leq1$ and let $0\leq b_{1}\leq b_{2}\leq1$.  Consider the rectangles $A=\{(x,y)\in\Omega\colon a_{1}\leq x\leq a_{2}\}$, $B=\{(x,y)\in\Omega\colon b_{1}\leq y\leq b_{2}\}$.  Show that the rectangles $A,B$ are independent.
\end{exercise}

\begin{exercise}
Let $\Omega=\R^{2}$ so that $\Omega\subset\R^{2}$.  Define a probability law $\P$ so that, for any set $A\subset\Omega$,
$$\P(A)=\frac{1}{2\pi}\iint_{A}e^{-(x^{2}+y^{2})/2}dxdy.$$
We previously verified that $\P(\Omega)=1$.  Let $-\infty<a_{1}\leq a_{2}<\infty$ and let $-\infty< b_{1}\leq b_{2}<\infty$.  Consider the infinite rectangles $A=\{(x,y)\in\Omega\colon a_{1}\leq x\leq a_{2}\}$, $B=\{(x,y)\in\Omega\colon b_{1}\leq y\leq b_{2}\}$.  Show that the rectangles $A,B$ are independent.
\end{exercise}

\begin{exercise}
Let $\Omega$ be a sample space and let $\P$ be a probability law on $\Omega$.  Let $A,B\subset\Omega$.  Assume that $A\subset B$.  Is it possible that $A$ is independent of $B$?  Justify your answer.
\end{exercise}


\begin{exercise}[\embolden{Inclusion-Exclusion Formula}]
In the Properties for Probability laws, we showed that $\P(A\cup B)=\P(A)+\P(B)-\P(A\cap B)$.  The following equality is a generalization of this fact.  Let $\Omega$ be a discrete sample space, and let $\P$ be a probability law on $\Omega$.  Prove the following.  Let $A_{1},\ldots,A_{n}\subset\Omega$.  Then:
\begin{flalign*}
\P(\cup_{i=1}^{n}A_{i})&=\sum_{i=1}^{n}\P(A_{i})-\sum_{1\leq i<j\leq n}\P(A_{i}\cap A_{j})+\sum_{1\leq i<j<k\leq n}\P(A_{i}\cap A_{j}\cap A_{k})\\
&\qquad\cdots+(-1)^{n+1}\P(A_{1}\cap\cdots\cap A_{n}).
\end{flalign*}
(Hint: begin with the identity $0=(1-1)^{m}=\sum_{k=0}^{m}(-1)^{k}\binom{m}{k}$, which follows from the Binomial Theorem.  That is, $1=\sum_{k=1}^{m}(-1)^{k+1}\binom{m}{k}$.  Now, let $x\in\Omega$ such that $x$ is in exactly $m$ of the sets $A_{1},\ldots,A_{n}$.  Compute the ``number of times'' that the element $x\in\Omega$ is counted for both sides of the Inclusion-Exclusion Formula.)
\end{exercise}

\begin{exercise}[\embolden{Derangements}]
\hfill
\begin{itemize}
\item Suppose you have a car with four tires, and the car mechanic removes all four tires.  Suppose the mechanic now puts the tires back on randomly, so that all arrangements of the tires are equally likely.  With what probability will no tire end up in its original position?  (Hint: let $A_{i}$ be the event that the $i^{th}$ tire is in the correct position, where $i=1,2,3,4$.  Then, use the Inclusion-Exclusion formula along with De Morgan's Law.)
\item Let $n$ be a positive integer.  Suppose your car has $n$ tires that are removed.  Suppose the mechanic now puts the tires back on randomly, so that all arrangements of the tires are equally likely.  With what probability will no tire end up in its original position?
\item Compute the latter probability as $n\to\infty$.
\end{itemize}
\end{exercise}

\begin{exercise}
Let $A,B,C$ be events that are pairwise independent, so that $\P(A\cap B)=\P(A)\P(B)$, $\P(A\cap C)=\P(A)\P(C)$, $\P(C\cap B)=\P(C)\P(B)$.  Assume that
$$\P(A)=1/2,\quad \P(B)=1/3,\quad \P(C)=1/4,\quad \P(A\cup B\cup C)=35/48.$$
Are the sets $A,B,C$ independent?  Explain.
\end{exercise}



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