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\title{Problem set 1 for Statistics 2211}
\author{B\'{a}lint Vir\'{a}g} \date{\today}
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\begin{center}
  {\LARGE \bf Statistics 2211 Assignment 1}

  Due Thursday, January 21 beginning of class
\end{center}
\ \\

\Prob{1} (a) Let $\mathcal F$ be a $\sigma$-field on a
countable set $\Omega$. Show that there exists pairwise
disjoint sets $\Omega_i,i\in I$ so that
$$
\mathcal F =\big\{\bigcup_{i\in J} \Omega_i\,:\, J \subset
I\}.
$$

(b) Give a counterexample to the above in the case $\Omega$
is not countable.

%\Prob{2} Let $\Var(X|\mathcal G)= \ev(X^2|\mathcal
%G)-\ev(X|\mathcal G)^2$. Show that
%$$
%\Var X= \ev \Var(X|\mathcal G) + \Var \ev (X|\mathcal G).
%$$
%
\Prob{2-9} Exercises 4.1.x for $x=2,3,4,5,6,7,8,9$ in
Durrett.

\Prob{10} Show that if for all $a\in \mathbb R$ we have 
$\ev|X-a|=\ev|\ev(X-a|\mathcal G)|$ then $X=\ev(X|\mathcal
G)$ a.s.


\end{document}
