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\begin{document}

\centerline {\bf Math 563 - Fall '14 - Homework 7} 

\medskip

\no {\bf Do 6 of the 7 problems}

\medskip

\no 1. (from Durrett) Thinning a process distribution. 
Let $N$ have a Poisson distribution with parameter $\lambda$. 
(Note $N$ is not a Poission process, just a single RV.) 
Let $X_n$ be an i.i.d. sequence taking only the values $0$ and $1$. 
Assume also that $N$ and $\{X_n\}_{n=1}^\infty$ are independent. 
Let 
$$
N_0 = | \{ n \le N: X_n=0\} |, \quad \quad
N_1 = | \{ n \le N: X_n=1\} |
$$ 
Here $| \quad |$ just means the cardinality of the set. Prove that 
$N_0$ and $N_1$ are independent, and each is Poisson with parameters 
$(1-p)\lambda$ and $p \lambda$ where $p=P(X_n=1)$. 

\medskip

\no 2. (from Durrett) Let $\Omega=\{a,b,c\}$. Given an example of 
a random variable $X$ and $\sigma$-fields $\field_1$,$\field_2$ such that 
$$
E[E[X|\field_1]|\field_2] \neq E[E[X|\field_2]|\field_1] 
$$

\medskip

\no 3. (from Durrett) Define $var[X|\field]$ to be 
$E[X^2|\field]-E[X|\field]^2$. Prove that 
$$
var(X) = E[var[X|\field]] + var(E[X|\field])
$$
(This is short). 

\medskip

\no 4. (from Durrett) Let $Y_n$ be an i.i.d. sequence of random variables.
Let $N$ be a random variable taking values in the positive integers which
is independent of $\{Y_n\}_{n=1}^\infty$. Let $\mu$ and $\sigma^2$ be the mean 
and variance of the $Y_n$. Let 
$$
X=\sum_{n=1}^N Y_n.
$$
So the number of terms in the sum is random. Show that 
$$
var(X) =  \sigma^2 E[N] + \mu^2 var(N) 
$$

\medskip

\no 5. Let $X_n$ be an independent sequence of random variables. 
Define a power series by 
$$
f(z)=\sum_{n=1}^\infty X_n z^n
$$ 
The radius of convergence of this power series depends on the $X_n$, so 
it is a random variable. Prove that this radius is equal to a constant
a.s.

\newpage

\no 6. (from Kallenburg) Let $X,Y,Z,W$
be random variables such that $(X,Y)$ has the same distribution as
$(Z,W)$. $X$ is integrable. 
Prove that $E[X|Y]$ has the same distribution as $E[Z|W]$.
The notation $E[X|Y]$ means $E[X|\sigma(Y)]$. 
Hint: recall problem 3 from the first homework which showed that 
any $\sigma(Y)$ measurable function can be written in the form 
$f(Y)$, where $f$ is a Borel-measurable function on the real line. 
So $E[X|Y]$ can be written in the form $f(Y)$. 
Show that $E[Z|W]$ is $f(W)$.

\medskip

\no 7. Let $X$ be a random variable with $E|X|<\infty$. 
In general the $\sigma$-field generated by $|X|$ will be smaller
than the $\sigma$-field generated by $X$. The goal of this problem is 
to find an explicit formula for $E[X| \sigma(|X|)]$ in a couple of 
special cases. 

\no (a) Suppose $X$ is a discrete RV (countable range). Find an explicit 
formula for $E[X| \sigma(|X|)]$.

\no (b) Now suppose the distribution of $X$ is absolutely continuous
with respect to Lebesgue measure and the density function is 
$f(x)$. Use your answer to part (a) to guess a formula for \
$E[X| \sigma(|X|)]$, and then show that your guess is correct. 
Your formula should involve $f$. 

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\end


\no 3. (from Durrett) Let $\field_1 \subset \field_2$ be $\sigma$-fields,
and $X$ a random variable with $E X^2 < \infty$. Prove that 
$$
E[(X-E[X|\field_2])^2] + E[(E[X|\field_2]-E[X|\field_1])^2] = 
E[(X-E[X|\field_1])^2]
$$

