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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 3, due October 7, 2020}

\bigskip\bigskip

\begin{enumerate} 

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

\begin{enumerate}
\item For fixed $a>0$, find a random variable $X$ such 
$$
\PP(|X|\ge a)=\frac{E(X^2)}{a^2}\,.
$$

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

\bigskip\item {\em The Hoeffding-Chernoff inequality
for fluctuations of size $t\sim \sqrt{n}$
(Panchenko 2.1.2)} \\
In the setting of Theorem 2.2 show that,
for $s>0$.
$$
\PP\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i-\mu)\ge s\right)
\le\exp\left(
-\frac{s^2}{2\mu(1-\mu)} + O\bigl(n^{-\frac12}\bigr)\right)\,.
$$

\bigskip\item {\em (Panchenko 2.1.5)} Suppose that
the random variables $X_1,\dots, X_n, Y_1,\dots, Y_n$ are independent
and, for all $i$, $X_i$ and $Y_i$ have the
same distribution.  Prove that
$$
\PP\left(\sum_{i=1}^n (X_i-Y_i)\ge 
\sqrt{2t\sum_{i=1}^n (X_i-Y_i)^2}\ \right)\le e^{-t}\,,\qquad
(t>0)\,.
$$
{\em Hint:} 
Think about a way to introduce
Bernoulli $B(\frac12)$ variables $\eps_i$ into the problem 
and then apply Hoeffding's inequality.
%(Take advantage of the fact that $Z_i=X_i-Y_i$ and
%$-Z_i$ have the same distribution.)




\bigskip\item {\em Continuous functions,
cont'd from last week's Problem 4 (Panchenko 1.3.2)}\\
Let $\mathcal{A}$ be the cylindrical $\sigma$-algebra
in $\RR^{[0,1]}$, and 
let $\Omega\subset [0,1]^\RR$ be the subset of all continuous functions
on $[0,1]$. Consider the $\sigma$-algebra 
$$\mathcal{B} := \{A\cap\Omega: A\in \mathcal{A}\}
$$ 
on $\Omega$, consisting of the continuous
functions in the cylindrical $\sigma$-algebra. 
Show that $\mathcal{B}$ contains the set
$$
C:= \left\{\omega\in\Omega: \int_0^1 \omega(t)\, dt <1\right\}\,.
$$ 
({\em Remark}. I switched the roles
of $\mathcal{A}$ and $\mathcal{B}$ --- makes
more sense to me this way.)

\bigskip
\item {\em Waiting for success}\\
Consider a sequence of independent tosses of a coin
that shows Heads with probability $p$, and Tails with
probability $q=1-p$. Let $X_i$ be indicator
that the $i$th toss comes up Heads.
Let $T_n$ be the number of the toss on
which Heads appears for the $n$-th time.


\begin{enumerate}
\item
{\em Geometric distribution}\quad
Find the distribution of $T_1$, and compute its expectation,
variance, and moment-generating function.

\item {\em Negative binomial distribution.}\quad
Write the distribution $\PP(T_n=t)$
in terms of suitable Binomial $(n,p)$ random variables.
(Do not try to simplify the formula.)

\item Show that $Y_1=T_1$ and $Y_n=T_n-T_{n-1}$, $n=2, 3,\ldots$ 
are all independent and identically distributed.  
Use this to compute the mean $\EE T_n$, the
variance $\Var(T_n)$, and the moment-generating
function $\EE e^{\lambda T_n}$.
\end{enumerate}

\bigskip\item
Consider again the repeated coin toss described
in the previous problem, and fix $n\ge 1$.
Let
$$
\sigma(T_n) = \{T_n^{-1}(A), A\subset\NN\}
$$
be the $\sigma$-algebra
generated by the random variable $T_k$.
For $j\ge 1$, find the conditional expectation $\EE(X_j\vert T_n)$.
(Express your answer as a random variable $f(T_n)$.)

\end{enumerate}
\end{document}

\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.

\end{document}


