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\begin{document}
\title{Stochastic Processes}
\author{Homework 2}
\date{Due in class Thursday, March 4}
\maketitle

\begin{enumerate}

\item Let $(\Omega, \F, P)$ be a probability space and let $\{A_i\}_{i
\in I}$ a partition of $\Omega$ into pairwise disjoint measurable
sets.  Prove \emph{Bayes' Theorem}, namely that
\[
  P(A_i | B) 
  = \frac{P(B | A_i) \, P(A_i)}%
         {\sum_{j \in I} P(B | A_j) \, P(A_j)}
\]
for every $i \in I$ and $B \in \F$.

\item Let $(\Omega, \F, P)$ be a probability space and let $\xi$ and
$\eta$ be independent random variables.  Let $f,g \colon \R \to \R$ be
Borel functions.  Show that $f(\xi)$ and $g(\eta)$ are independent.

(Recall that $f$ is a Borel function if $f^{-1}(B)$ is a Borel set for
every Borel set $B$.)

\item  Let $(\Omega, \F, P)$ be a probability space, $\xi$ a random
variable, and $\eta$ a discrete random variable taking values $\{ y_i
\}_{i \in I}$ where $I$ is at most countable.  Show that
\[
  E(\xi) = \sum_{i \in I} E(\xi | \eta^{-1}\{y_i\}) \, 
  P \eta^{-1}\{y_i\} \,.
\]

\item Two dice are tossed.  What is the conditional expectation
$E(\xi|\eta)$ of the total amount $\xi$ shown given $\eta$, the
absolute value of the difference of the amounts shown?

\item Let $\Omega = [-1/2,1/2]$ with the Lebesgue measure.  Find the
conditional expectation $E(\xi|\eta)$ if
\[
  \xi(x) = x \,, \qquad \eta(x) = x^2 \,.
\]

\end{enumerate}

\end{document}
