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\begin{document}
\begin{center}
{\bf Probability 1\\
CEU Budapest, fall semester 2013}\\
Imre Péter Tóth\\
{\bf Homework sheet 1 -- due on 23.09.2013}\\
\end{center}


\begin{enumerate}
%
 \item Define a $\sigma$-algebra as follows:
  \begin{definition}
   For a nonempty set $\Omega$, a family $\sigalg$ of subsets of $\omega$ (i.e. $\sigalg\subset 2^\Omega$, where $2^\Omega:=\{A:A\subset\Omega\}$ is the power set of $\Omega$) is called a $\sigma$-algebra over $\Omega$ if
    \begin{itemize}
     \item $\emptyset\in\sigalg$
     \item if $A\in \sigalg$, then $A^C:=\Omega\setminus A\in \sigalg$ (that is, $\sigalg$ is closed under complement taking)
     \item if $A_1,A_2,\dots\in\sigalg$, then $(\cup_{i=1}^\infty A_i)\in \sigalg$ (that is, $\sigalg$ is closed under countable union).
    \end{itemize}
  \end{definition}
Show from this definition that a $\sigma$-algebra is closed under countable intersection, and under finite union and intersection.
%
 \item
  \begin{enumerate}
   \item We toss a biased coin, on which the probability of heads is some $0\le p \le 1$. Define the random variable $\xi$ as the indicator function of tossing heads, that is
    $$
      \xi:=\begin{cases}
            0, \text{ if tails}\\
            1, \text{ if heads}
            \end{cases}.
    $$
    \begin{enumerate}
     \item Describe the distribution of $\xi$ (called the Bernoulli distribution with parameter $p$) in the ``classical'' way, listing possible values and their probabilities,
     \item and also by describing the distirbution as a measure on $\IR$, giving the weight $\prob(\xi\in B)$ of every Borel subset $B$ of $\IR$.
     \item Calculate the expectation of $\xi$.
    \end{enumerate}
   \item We toss the previous biased coin $n$ times, and denote by $X$ the \emph{number of heads} tossed.
    \begin{enumerate}
     \item Describe the distribution of $X$ (called the Binomial distribution with parameters $(n,p)$) by listing possible values and their probabilities.
     \item Calculate the expectation of $X$ by integration (actually summation in this case) using its distribution,
     \item and also by noticing that $X=\xi_1+\xi_2+\dots +\xi_n$, where $\xi_i$ is the indicator of the $i$-th toss being heads, and using linearity of the expectation.
    \end{enumerate}
   \end{enumerate}
%
 \item \emph{The Fatou lemma} is the following
  \begin{theorem}
   Let $(\Omega,\sigalg,\mu)$ be a measure space and $f_1$, $f_2$, \dots a sequence of measureabale functions $f_n:\Omega\to\IR$, which are nonneagtive, e.g. $f_n(x)\ge 0$ for every $n=1,2,\dots$ and every $x\in\Omega$. Then
    $$\int_\Omega \liminf_{n\to\infty} f_n(x) \diff \mu(x)\le \liminf_{n\to\infty} \int_\Omega f_n(x)\diff \mu(x)$$
   (and both sides make sense).
  \end{theorem}
 Show that the inequality in the opposite direction is in general false, by choosing $\Omega=\IR$, $\mu$ as the Lebesgue measure on $\IR$, and constructing a sequence of nonnegative $f_n:\IR\to\IR$ for which $f_n(x)\xrightarrow{n\to\infty} 0$ for every $x\in\IR$, but $\int_\IR f_n(x) \diff x\ge 1$ for all $n$.
%
 \item The \emph{ternary} number $0.a_1 a_2 a_3 \dots$ is the analogue of the usual decimal fraction, but writing numbers in base $3$. That is, for any sequence $a_1,a_2,a_3,\dots$ with $a_n\in\{0,1,2\}$, by definition
 $$0.a_1 a_2 a_3 \dots:=\sum_{n=1}^\infty \frac{a_n}{3^n}.$$
 Now let us construct the ternary fraction form of a random real number $X$ via a sequence of fair coin tosses, such that we rule out the digit $1$. That is,
 $$a_n:=\begin{cases}
        0, \text{ if the $n$-th toss is tails,}\\
        2, \text{ if the $n$-th toss is heads}\\
       \end{cases},
 $$
 and setting $X=0.a_1 a_2 a_3 \dots$ (ternary). In this way, $X$ is a ``uniformly'' chosen random point of the famous \emph{middle-third Cantor set} $C$ defined as
 $$C:=\left\{\sum_{n=1}^\infty \frac{a_n}{3^n}\, ,\, a_n\in\{0,2\}\, (n=1,2,\dots)\right\}.$$ 
 Show that
  \begin{enumerate}
   \item The distribution of $X$ gives zero weight to every point -- that is, $\prob(X=x)=0$ for every $x\in\IR$. (As a consequence, the cumulative distribution function of $X$ is continuous.)
   \item The distribution of $X$ is not absolutely continuous w.r.t the Lebesgue measure on $\IR$.
  \end{enumerate}
%
\end{enumerate}

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