\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,times}

\advance\voffset -2cm
\advance\hoffset -2cm
 
\textwidth   6.5in
\textheight  8.5in

\def\EE {{\mathbb E}}
\def\PP {{\mathbb P}}
\def\RR {{\mathbb R}}

\begin{document}

\section*{MAT 1600 : Probability I\\
Assignment 1, due September 23, 2020}

\bigskip\bigskip

\begin{enumerate} 

\bigskip \item {\em Monotone limits}\\
Let $\mathcal{A}$ be an algebra.
Suppose that  $\mathcal{A}$ is closed under
countable {\em increasing} unions, i.e.,
$\bigcup _{j=1}^\infty B_j\in \mathcal{A}$
whenever $B_j\in \mathcal{A}$ and $B_j\subset B_{j+1}$
for each $j=1,2,\dots$.

Prove that $\mathcal{A}$ is a $\sigma$-algebra, i.e., $\mathcal{A}$
is in fact closed under general countable unions.

\bigskip \item {\em (Durrett 1.1.6)} \quad
A set $A\subset\{1,2,\dots\}$
is said to have {\bf asymptotic density} $\theta$ if
$$
\lim_{n\to\infty} \frac1n \left\vert A\cap \{1,2,\dots, n\}\right\vert =
\theta\,.
$$
Let $\mathcal{A}$ be the collection of sets for which the asymptotic
density exists. 

(a) Is $\mathcal{A}$ a $\sigma$-algebra? an algebra?

(b) Is $\theta$ additive? $\sigma$-additive on $\mathcal{A}$?

\bigskip \item {\em Section property (Lieb \& Loss, p. 8)}
\begin{enumerate}
\item Let $A$ be a Borel set in $\RR^2$. Prove that, for every
$y\in\RR$, the cross section
$$
A(y)=\{x\in\RR: (x,y)\in A\}
$$
is a Borel set. \ ({\em Hint:} Consider the collection
of sets whose cross sections are Borel, and use Dynkin's theorem.)
\item The corresponding statement fails for Lebesgue measurable
sets. Briefly explain why!
\end{enumerate}

\bigskip\item {\em Tail sum formula (b-Panchenko,
Problem 1.2.5)}\\
For any nonnegative random variable $X\ge 0$,
and any $p>0$,
prove that
$$
\EE(X^n)= \int_0^\infty pt^{p-1}\, \PP(X>t)\, dt\,.
$$

\newpage

\bigskip\item {\em Uniformization}
\begin{enumerate}
\item {\em (Panchenko 1.2.1)}\quad
If a random variable $X$ has continuous c.d.f. $F(t)$, show that
the random variable $F(X)$ is uniform on $[0,1]$,
i.e., the law of $F(X)$ is Lebesgue measure on $[0,1]$.
\item {\em (Panchenko 1.2.2)}\quad
Conclude that $\int F(t)\, dF(t)=\frac12$.
\end{enumerate}



\bigskip\item (a) {\em Inclusion-exclusion (Durrett 1.6.9)}\\
For any collection of events
$A_1,\dots, A_n$, show that
$$
\PP\left( \bigcup_{i=1}^n A_i\right)
= \sum_{\emptyset \ne F\subset\{1,\dots, n\}}
(-1)^{|F|+1}
\PP\left(\bigcap_{i\in F} A_i\right)\,.
$$
Here, $|F|$ denotes the number of elements of the set $F$.

{\em Hint:} Expand the indicator function 
$$
I_{\bigcup A_i} = 1-\prod\left(1-I_{A_i}\right)
$$
and take expectations.  (Induction over $n$ is a hassle.)

(b) {\em The forgetful secretary} \\
Consider the uniform probability measure
on $S_n$, the set of permutations
of $\{1,\dots, n\}$. Find the probability that
a random permutation has no fixed point,
$$
p_n := \PP\left(\pi(i)\ne i\ \mbox{for all}\ i=1,\dots,n\right)\,,
$$
and compute $p=\lim p_n$.
\ ({\em Hint:} Take $A_i=\{\pi\in S_n: \pi(i)=i\}$.)

\end{enumerate}
\end{document}

\item (a) {\em Inclusion-exclusion}\\
Show that $\PP(A\cup B)=\PP(A) + \PP(B)-\PP(A\cap B)$ 
for any pair of events $A$, $B$.

(b) {\em Conditional probability}\\
Given an event $B$ with $\PP(B)\ne 0$, define
$$
\PP(A\vert B) = \frac{\PP(A\cap B)}{\PP(B)}\,.
$$
Show that $\PP(\cdot \vert B)$ defines a probability
measure (on which probability space?)

\bigskip\item {\em (Durrett 1.6.14)} Let $X\ge 0$ be a 
nonnegative random variable. Without assuming
that $B(1/X)<\infty$, show that
$$
\lim_{y\to\infty} yB(1/X;X>y) =0\,,\qquad
\lim_{y\to 0} yB(1/X;X>y) =0\,.
$$


\bigskip\item {\em The Vitali-Hahn-Saks theorem}\\
Let $(\PP_n)_{n\ge 1}$ be a sequence of probability
measures on $(\Omega,\mathcal{F})$.
Suppose that 
$$ m(A):= \lim_{n\to\infty} \PP_n(A)
$$
exists for each $A\in\mathcal{\mathcal F}$. Prove that
$m$ is a measure.

\bigskip\item {\em Chebyshev's inequality is and is not sharp
(Durrett 1.6.2)} 

(a) For fixed $a>0$, find a random
variable $X$ such that Eq. (1.6.1) holds with equality,
$$
\PP(|X|\ge a)=\frac{E(X^2)}{a^2}\,.
$$

(b) On the other hand, if $X$ is a random variable with $0<E(X^2)<\infty$, 
then
$$
\lim_{a\to \infty} a^2\PP(|X|\ge a)=0\,.
$$

\bigskip\item Use Jensen's inequality to give an alternative prove of
H\"older's inequality: If $1<p,q<\infty$
with $\frac1p +\frac1q=1$ are H\"older dual exponents, then
$$
\EE|XY| \le ||X||_p\,||Y||_q\,,
$$
in particular, the left hand side is finite whenever the right hand side is.

{\em Hint:}
In the special case $||Y||_q=1$, use $Y$ to define
a probability measure; then rescale to
get the general statement. 

\bigskip\item {\em (Durrett 1.2.3)} \quad Show that a distribution function 
$F(x)=\PP(X\le x)$ has at most countably many discontinuities.

\bigskip\item 
(a) {\em Change of variables (Durrett 1.2.5)} \\
Suppose $X$ has continuous density $f$,
that $\PP(\alpha\le X\le \beta)=1$, and that $g$ is a strictly increasing,
differentiable function on $(\alpha,\beta)$. Show that $Y=g(X)$ has density
$$
\left\{
\begin{array}{ll}
\frac{f(g^{-1}(y))}{g'(g^{-1}(y))}\,,\quad & \text{if}\ g(\alpha)<y<g(\beta)\,,
\\
0\,,& \text{otherwise}\,.
\end{array}\right.
$$
In particular, when $g(x)=ax+b$ with $a>0$, then the density of
$Y$ is $\frac1a f\left(\frac{y-b}{a}\right)$.

(b) {\em Log-normal distribution (Durrett 1.2.6)}\\
Find the density of $\exp(X)$ when $X$ has a standard normal distribution.



\bigskip\item 
(a)  {\em (Durrett 1.2.7)} \quad Suppose $X$ has density function
$f$. Compute the distribution function of $Y=X^2$
and then differentiate it to find its density function.

(b) {\em Chi-square distribution}\\
Find the density of $X^2$ when $X$ has a standard normal
distribution.

\bigskip\item {\em (Durrett 1.3.8)} 
Let $X,Y$ be random variables,
and let $\sigma(X)$ be the $\sigma$-algebragenerated by $X$.
Show that $Y$ is measurable with respect to $\sigma(X)$, if and
only if $Y=f(X)$ for some measurable function $f:\RR\to\RR$.
({\em Hint:} First consider the case where $Y$ is a simple function.)

\end{enumerate}

\end{document}

