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\def\eps{\varepsilon}
\def\EE {{\mathbb E}}
\def\NN {{\mathbb N}}
\def\PP {{\mathbb P}}
\def\RR {{\mathbb R}}
\def\Var {\mathrm {Var}\,}

\begin{document}

\section*{MAT 1600 : Probability I\\
Assignment 4, due October 14, 2020}

\bigskip\bigskip

\begin{enumerate} 

\bigskip 
\item {\em Panchenko 2.2.1} \quad Let $(X_n)_{n\ge 1}$ be i.i.d. random 
variables with  $\EE|X_i|^p<\infty$ for some $p>0$. Show that
$ \max_{i\le n} n^{-\frac1p}|X_i|\to 0$
in probability as $n\to\infty$.

\bigskip\item {\em (Probabilistic proofs
of the Borel-Cantelli Lemma (2.2.4)}\\
Let $(A_n)_{n\ge a}$ be a sequence of events,
and let $N$ be the number of events that occur
(i.e., $N(\omega)=\#\{n\ge 1: \omega\in A_n\}$).

\begin{enumerate}
\item If $A_n$ is a sequence of 
events with $\sum_{n\ge 1}\PP(A_n)<\infty$, then $\PP(N<\infty)=1$.\\
(Consider the expectation of $N$.)
\item If the events are independent and 
$\sum_{n\ge 1} \PP(A_n)=\infty$, then $\PP(N<\infty)=0$.\\
(You can start, for example, by showing that
$\PP(N=0)=0$.)
\end{enumerate}

\bigskip\item {\em Weak LLN for $U$-statistics (Panchenko 2.2.2)}
\begin{enumerate}
\item Look up $U$-Statistic on Wikipedia.
\item If $(X_n)_{n\ge 0}$ are i.i.d. with mean $\mu$ and varianc e
$\sigma^2<\infty$, show that
$$
{n\choose 2}^{-1} \sum_{1\le i<j\le n} X_iX_j\to\mu^2
$$
in probability as $n\to\infty$.
\end{enumerate}

\bigskip\item {\em (Panchenko  2.2.3)} If $u:[0,1]^n\to \RR$
is continuous, prove that
$$
\sum_{0\le j_1, \ldots, j_k\le n}
u\left(\frac{j_1}{n}, \ldots, \frac{j_k}{n}\right)
\prod_{i\le k} {n\choose j_i}x_i^{j_i}(1-x_i)^{n-j_i}
\  \longrightarrow \ u(x_1,\dots, x_n)
$$
as $n\to\infty$, uniformly on $[0,1]^k$.

\bigskip \item {\em (Panchenko 2.1.7)} Suppose that the chances of 
winning the jackpot in a lottery is 1:139,000,000. Assuming that
100,000,000 people played independently of each other, estimate the
probability that 3 of them will have to share the jackpot. Give a bound on 
the quality of your estimate.

\bigskip 
\item {\em Monte Carlo integration (Durrett 2.2.3)} \quad Let $f$ be
a Borel measurable, Lebesgue integrable  function on the unit interval 
$[0,1]$. The objective is to construct a
probabilistic method for computing the integral
$$
I =\int_0^1 f(x)\, dx\,.
$$

Let $U_1,U_2,\dots$ be independent and uniformly distributed
on $[0,1]$, and let
$$
I_n= \frac1n \bigl(f(U_1)+\cdots+ f(U_n)\bigr)\,
$$
be average of the first $n$ values.

(a) Show that $I_n\to I$ in probability.

(b) If, moreover, $\int|f(x)|^2\, dx <\infty$,
use Chebyshev's inequality to estimate 
$$P(|I_n-I|)>an^{1/2}\,,\quad \text{for}\  a>0\,.
$$

{\em Remark.} This method for computing integrals
can be adapted (by change of variables)
to numerically integrate functions of many variables 
over complicated regions.

\bigskip
\end{enumerate} 
\end{document}

\bigskip
\item {\em (Durrett 2.2.5)} Let $X_1,X_2,\dots$ be i.i.d. with
$P(X_i>x)= \frac{e}{x}\log x$
 for $x\ge e$. 

\begin{enumerate}
\item Show that $\EE X_i=\infty$.

\item Construct a sequence of constants
$\mu_n\to\infty$ such that $S_n/n-\mu_n\to 0$ in probability..

\end{enumerate}
\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
$$
\EE|XY| \le \left(\EE|X|^p\right)^\frac1p \left(\EE|Y|\right)^\frac1q\,,
$$
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 
(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 1.2.3)} \quad Show that a distribution function 
$F(x)=\PP(X\le x)$ has at most countably many discontinuities.


