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\centerline {\bf Math 563 - Fall '21 - Midterm (take home)} 

\medskip

\noindent {\bf \large Do 4 of the 5 problems. Do not turn in more than 4.}

\medskip

\noindent {\bf Due Wednesday, Oct 27, 11:59 pm - no late papers accepted}

\smallskip

\no {\bf The fine print: } You are supposed to do this exam on your
own. This means you should not talk to anyone about the exam.
You can ask me questions, but I will only answer questions about
what the problem is asking, not about how to do it. 
You can consult your class notes, past homeworks and Durrett.
It is probably possible to find the solution to every one of these problems 
somewhere on the web. Don't do this!

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Borel-Cantelli
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\no 1. Let $X_n$ be an independent, identically distributed 
sequence of non-negative random variables. 
Prove that $EX_1 <\infty$ if and only 
if $P(X_n \ge n \, \, i.o.)=0$. 

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% standard maching, conditional prob. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\no 2. Let $(\Omega,\f,P)$ be a probability space. 
Let $A$ be an event with $P(A)>0$. 
Define a new probability measure on $(\Omega,\f)$ by 
$Q(B)=P(A \cap B) / P(A)$. (So $Q(B)=P(B|A)$.) 
Let $X$ be a nonnegative random variable 
which is independent of $A$, meaning that $\sigma(X)$ and $\sigma(A)$ 
are independent. ($\sigma(A)$ is just $\{A,A^c,\Omega,\emptyset\}$.)
Prove that 
\be
\int X dP = \int X dQ
\label{eqq}
\ee
Hint: start with a simple random variable $X$.

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% law of large numbers  and CLT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\no 3. Let $X_n$ be an i.i.d. sequence with zero mean and finite variance. 
Prove that
$$
\frac{\sum_{k=1}^n X_k}{\left( \sum_{k=1}^n X_k^2 \right)^{1/2}}
$$
converges in distribution to a standard normal. 
Hint: what does the strong law of large numbers tell you about the
denominator?

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% convergence in distribution and char func
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\no 4. The random variables $X_n,Y_n,X,Y$ are all defined on the same
probability space. Suppose that $X_n$ converges to $X$ in distribution and
that $Y_n$ converges to $Y$ in distribution. 
Suppose also that $X$ and $Y$ are independent
and that for each $n$ the RV's $X_n$ and $Y_n$ are independent.
Prove that $X_n+Y_n$ converges in distribution to $X+Y$. 
Hint: characteristic functions. 

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% joint distribution 
% independence 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

\no 5. Let $(\Omega,\field,P)$ be a probability space and suppose that 
$\Omega$ is countable. Let $X_n$ be a sequence of random variables. 
Prove that $X_n \rightarrow 0$ a.s. if and only if 
$X_n \rightarrow 0$ in probability.




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\no 5. Let $X_1,X_2$ be random variables. Suppose that for all
bounded measurable functions $f(x,y)$ from $\reals^2$ to $\reals$ we
have
\bea
E[f(X_1,X_2)] = \frac{1}{2} \int_0^1 \, f(x,0) \, dx
+ \frac{1}{2} \int_0^1 \, f(x,1) \, dx
\eea
where $dx$ is integration with respect to one-dimensional Lebesgue measure. 

\no (a) Find the joint distribution measure $\mu_{X_1,X_2}$ and
the individual distribution measures, $\mu_{X_1}$ and $\mu_{X_2}$.

\smallskip
\no (b) Are $X$ and $Y$ independent?

