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\begin{document}

\section*{MAT 1600 : Probability I\\
Assignment 2, due September 30, 2020}

\bigskip\bigskip

\begin{enumerate} 

\bigskip\item {\em (Durrett 2.1.11)} Find four random
variables taking values in $\{-1,0, 1\}$
so that any three are independent but all four are not.
\quad ({\em Hint:} Consider products of
independent random variables.)

\bigskip\item {\em The Vitali-Hahn-Saks theorem}\\
Let $(\PP_n)_{n\ge 1}$ be a sequence of probability
measures on $(\Omega,\mathcal{A})$.
Suppose that 
$$ m(A):= \lim_{n\to\infty} \PP_n(A)
$$
exists for each $A\in\mathcal{\mathcal A}$. Prove that
$m$ is a measure.
{\em Remark:} In this generality, the theorem
is hard (though it has an elementary proof). I will post an
outline on Friday.

\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.)

\bigskip\item {\em Continuous functions (Panchenko 1.3.1)}\\
Does the set $C([0,1])$ belong to the cylindrical $\sigma$-algebra
$\mathcal{A}$ on $\RR^{[0,1]}$?\\
({\em Panchenko's hint:} An exercise in Section 1
might be helpful.)

\bigskip\item {\em Uncorrelated but not independent
(Durrett 2.1.7)}
Consider $\Omega=(0,1)$, with the standard
Borel $\sigma$-algebra and Lebesgue measure.
For positive integers $n\ne m$,
show that the
random variables $X(\omega)=\sin(2\pi n\omega)$
and $Y(\omega)=\sin(2\pi m\omega)$
satisfy $E(XY)=EX=EY=0$. But $X$ and $Y$ are not
independent.

\bigskip\item Let $X\sim {\rm Poiss}(\lambda)$
and $Y\sim {\rm Poiss}(\mu)$ be independent random variables
with Poisson distribution, as defined in Example 1.1.1.
\begin{enumerate} 
\item Determine the distribution of $Z=X+Y$.  
\end{enumerate}

Conversely, let $N\sim {\rm Poiss}(\lambda)$ 
be a Poisson-distributed random variable.
If $N=n$, toss a biased coin $n$ times
and let $N_1$ be the number of Heads you get.\\
Then $P(N_1=k\big\vert N=n)= {n\choose k} p^k(1-p)^{n-k}$,
where $p$ is the bias. 
\begin{enumerate}
\setcounter{enumii}{1}
\item Argue that $N_0=N-N_1$ and $N_1$ are
independent, and determine their distribution.
(Avoid excessive computation.)
\end{enumerate}
\end{enumerate}


\end{document}

\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.

\end{document}
\bigskip\item {\em Continuous functions, cont'd 
(Panchenko 1.3.2)}\\
Let $\mathcal{SA}$ be the cylindreical $\sigma$-algebra
in $\RR^{[0,1]}$, and 
let $\omega$ be the subset of all continuous functions
on $[0,1]$. Consider the $\sigma$-algebra 
$$\mathcal{B} := \{A\cap\Omega: A\in \mathcal{A}\}
$$ 
on $\Omega$.  Show that $\mathcal{B}$
contains the set
$$
B_1:= \left\{\omega\in\Omega: \int_0^1 \omega(t)\, dt <1\right\}\,.
$$ 
({\em Remark}. I switched the roles
of $\mathcal{A}$ and $\mathcal{B}$ --- makes
more sense to me this way.)


