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\title{Stochastic Processes}
\author{Summer Semester 2026, Exercise 4}
\date{Due Thursday, June 4, 2026}
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\begin{enumerate}

\item Let $\xi_i$, $i=1,2$ be independent Bernoulli-distributed random
variables with parameter $p \in [0,1]$.
\begin{enumerate}
\item Set $\eta = 4 \, \xi_1 \, \xi_2$.  Find
\[
  \mathbb E[\eta | \xi_1] \,.
\]

\item Set $\zeta = \boldsymbol{1}_{\{\xi_1 + \xi_2 = 0\}}$.  Find
\[
  \mathbb E[\xi_i | \zeta] \,.
\]

\item Are $\mathbb E[\xi_1 | \zeta]$ and $\mathbb E[\xi_2 | \zeta]$
independent?
\end{enumerate}


  
\item Let $\xi_1, \dots, \xi_n$ be independent Bernoulli-distributed
random variables with parameter $p \in [0,1]$, and set
\[
  \eta_n = \xi_1 + \dots + \xi_n \,.
\]
Compute
\[
   \mathbb E [\eta_{n+1} | \xi_1, \dots, \xi_n] \,.
\]

\item Let $(\Omega, \mathcal F, P)$ be a probability space and
$\mathcal G \subset \mathcal F$ a sub-$\sigma$-algebra.  Let $\xi$ be
a random variable with finite variance and define
\[
  \eta = \mathbb E[\xi | \mathcal G] \,.
\]
Prove that if $\mathbb E[\xi^2] = \mathbb E[\eta^2]$, then
$\xi=\eta$ a.s.


\end{enumerate}
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