Basic Concepts of Probability Theory

A fair coin is tossed until heads comes up two times in a row. Construct a probability space. Find the probability that the number of tosses does not exceed 5.

A fair coin is tossed until the same side comes up two times in a row. Construct a probability space. Find the probability that the number of tosses will be even.

A fair coin is tossed until heads appears $r$ times. Construct a probability space. How many elementary events does the event ``the experiment ends after the $n$-th toss'' contain?

A point is chosen at random on the segment [ 0,1 ]. Let the event $A_{n}$ mean that the point is chosen from the half-interval $\left[0, \frac{n}{n+1}\right)$, and the event $B_{n}$ that the point is chosen from the interval $\left(0, \frac{1}{n}\right)$. What do the events $\bigcup_{n = 1}^{\infty } A_{n}$ and $\bigcap_{n = 1}^{\infty } B_{n}$ mean?

Prove that every algebra is a semialgebra.

Give an example of a semialgebra that is not an algebra.

Give an example of an algebra that is not a $\sigma$-algebra.

Let $\Omega = \left\{ 0,1,2\right\}$. Give examples of $\sigma$-algebras containing the sets $A = \left\{ 0,1\right\}$ and $B = \left\{ 1,2\right\}$.

Describe the $\sigma$-algebra of subsets of the segment $[0,1]$ generated by the sets:

Let $\Omega$ be an uncountable set. Describe the $\sigma$-algebra generated by:

Prove that every finite $\sigma$-algebra of subsets of a space $\Omega$ is generated by some finite partition of $\Omega$.

Let $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ be two $\sigma$-algebras of subsets of a space $\Omega$. Are the following classes of sets $\sigma$-algebras:

Let $A_{1}, A_{2}, \ldots$ be a sequence of disjoint subsets of a space $\Omega$. Determine the cardinality of the $\sigma$-algebra generated by this sequence.

Prove that if $\mathscr {A}_{1}, \mathscr {A}_{2}, \ldots$ is a nondecreasing sequence of $\sigma$-algebras, then $\mathscr {A} = \bigcup_{n = 1}^{\infty } \mathscr {A}_{n}$ is an algebra.

Can the total number of events of some probability space be equal to $129 ; 130 ; 128$ ?

Let $\mathscr {A}$ be a $\sigma$-algebra of subsets of a space $\Omega$. Prove that if $\mathscr {A}$ is infinite, then there exists a countable sequence of nonempty disjoint elements of $\mathscr {A}$.

Prove that for any space $\Omega$, no $\sigma$-algebra of its subsets can have countable cardinality.

Can the number of elementary events be strictly greater than the number of all events?

Can it happen that:

The number of elementary events of some probability space equals $n$. Indicate the minimum and maximum possible values for the number of events.

Describe the $\sigma$-algebra generated by:

Does the set of all events whose probabilities are expressed by rational numbers form a $\sigma$-algebra?

Prove that:

Let $A_{1}, A_{2}, \ldots$ be a sequence of events. Prove that the events $\limsup A_{n}$ and $\liminf A_{n}$ belong to the $\sigma$-algebra generated by this sequence.

Prove the following relations:

$\lim \sup A_{n} \cap \lim \inf B_{n} \subset \lim \sup \left(A_{n} \cap B_{n}\right) \subset \lim \sup A_{n} \cap \lim \sup B_{n}$.

Let $A_{1} \supset A_{2} \supset \ldots$ be a nonincreasing sequence of events. Prove that

Let $A_{1} \subset A_{2} \subset \ldots$ be a nondecreasing sequence of events. Prove that

Prove that

Let ($\Omega , \mathscr {A}, P$) be an arbitrary probability space. Prove that the set of values of the function $\mathbb {P}\left(A\right), A \in \mathscr {A}$ is a closed subset of the segment $[0,1]$.

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a nonatomic probability space. Prove that the set of values of the function $\mathbb {P}\left(A\right)$ is the entire segment $[0,1]$.

Let ($\Omega , \mathscr {A}, P$) be a probability space. Call events $A$ and $B$ equivalent if $\mathbb {P}\left(A \Delta B\right) = 0$. Let $\mathfrak {B}$ be the set of equivalence classes of events. For any two equivalence classes of events $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ set

where $B_{1}$ and $B_{2}$ are arbitrary representatives of the classes $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$, respectively. Prove that $\rho$ is a metric on $\mathscr {F}$ and that $\mathscr {B}$, with the metric so introduced, is a complete metric space.

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space, where $\mathscr {A}$ is generated by some algebra $\mathscr {B}$. Prove that for any $\varepsilon > 0$ and any $A \in \mathscr {A}$ there exists $B \in \mathscr {B}$ such that $\mathbb {P}\left(A \Delta B\right) \leqslant \varepsilon$.

Give an example of a sequence of events $A_{1}, A_{2}, \ldots$ such that $\mathbb {P}\left(A_{n}\right) \rightarrow 1$ as $n \rightarrow \infty$, but

for any $n$.

Prove that for any sequence of events $A_{1}, A_{2}, \ldots$

Are the following relations valid:

Let $A_{1}, A_{2}, \ldots$ be a sequence of events such that

Prove that $\mathbb {P}\left(\limsup A_{n}\right) = 0$.

Let $A_{1}, A_{2}, \ldots$ be a sequence of events. Is it true that if $\liminf \mathbb {P}\left(A_{n}\right) > 0$, then $\mathbb {P}\left(\lim \inf A_{n}\right) > 0$?

Let $A_{1}, A_{2}, \ldots$ and $B_{1}, B_{2}, \ldots$ be two sequences of events, with $\mathbb {P}\left(B_{n}\right) \rightarrow 1$ as $n \rightarrow \infty$. Prove that

provided that at least one of the indicated limits exists.

Let $A_{4}, A_{2}, \ldots$ and $B_{1}, B_{2}, \ldots$ be two sequences of events, with $\mathbb {P}\left(B_{n}\right) \rightarrow 1$ as $n \rightarrow \infty$. Prove that if

then

Can condition (1) be dropped?

Let $\xi$ and $\eta$ be random variables defined on the same probability space ($\Omega , \mathscr {A}, \mathbb {P}$). Prove that the sets

are events.

Let $(\Omega , \mathscr {A}, \mathbb {P})$ be a probability space, and $\xi (\omega )$ a real-valued function defined on $\Omega$ such that for every real $c$ the set

is an event. Must $\xi$ be a random variable?

Let $\xi$ and $\eta$ be random variables defined on the same probability space. Prove that the following functions are random variables:

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space, and $\xi (\omega )$ a real-valued function defined on $\Omega$. Must $\xi$ be a random variable if the following is a random variable:

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables defined on the same probability space. Prove that the functions $\inf \xi_{n}$ and $\sup \xi_{n}$ are random variables.

Let $\mathfrak {M}$ be an uncountable family of random variables. Must the functions $\inf \xi$ and $\sup \xi$ be random variables?

Prove that if $\xi$ is a random variable and $f(x)$ is a Borel function, then $\eta = f(\xi )$ is a random variable.

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables defined on the same probability space, and $\varphi \left(x_{1}, \ldots , x_{n}\right)$ a real-valued Borel function of $n$ variables. Prove that the function $\varphi \left(\xi_{1}, \ldots , \xi_{n}\right)$ is a random variable.

Let $A$ and $B$ be events of the same probability space, and $I_{A}$ and $I_{B}$ their indicators. Prove that

An urn contains 3 white and 2 black balls. The experiment consists of drawing all the balls from the urn in succession. Construct a probability space. Describe the $\sigma$-algebra generated by the random variable $\xi$, if:

The probability space ($\Omega , \mathscr {A}, P$) is the segment $[0,1]$ with the $\sigma$-algebra of Borel subsets and Lebesgue measure. Describe the $\sigma$-algebra generated by the random variable $\xi$, if:

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space, where $\Omega$ is the real line and $\mathscr {A}$ is the $\sigma$-algebra of Borel sets. Describe the $\sigma$-algebra generated by the random variable $\xi = \cos \omega$.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables defined on the same probability space. Prove that the following sets are events:

Give an example of a nontrivial probability space ($\Omega , \mathscr {A}, P$) and a random variable $\xi$ on it, such that any random variable $\eta$ defined on this probability space is a Borel function of $\xi$.

On the probability space ($\Omega , \mathscr {A}, \mathbb {P}$), which is the segment $[0,1]$ with the $\sigma$-algebra of Borel subsets and Lebesgue measure, a sequence of random variables $\xi_{1}, \xi_{2}, \ldots$ is defined as follows: $\xi_{n} = \omega^{n}$. Set $A_{n} = \left\{ \omega \in \Omega : \xi_{n} \leqslant 1 / n\right\}$. Find $\bigcup_{n = 1}^{\infty } A_{n}$ and $\bigcap_{n = 1}^{\infty } A_{n}$.

Let $\xi$ and $\eta$ be equivalent random variables, that is, $\mathbb {P}\left(\xi \neq \eta \right) = 0$. Prove that if $\mathbb {E}\left[\xi \right]$ exists, then $\mathbb {E}\left[\eta \right]$ exists and $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\eta \right]$.

Prove that $\mathbb {E}\left[\xi^{2}\right] = 0$ is equivalent to $\mathbb {P}\left(\xi = 0\right) = 1$.

Let $\xi$ and $\eta$ be random variables. Prove that if $\mathbb {E}\left[\xi \right]$ and $\mathbb {E}\left[\eta \right]$ exist, then $\mathbb {E}\left[\max \left\{ \xi , \eta \right\} \right]$ exists. Is the converse true?

Let $\xi$ and $\eta$ be random variables. Prove that if $\mathbb {E}\left[\max \left\{ \xi , \eta \right\} \right]$ and $\mathbb {E}\left[\min \left\{ \xi , \eta \right\} \right]$ exist, then $\mathbb {E}\left[\xi \right]$ and $\mathbb {E}\left[\eta \right]$ exist, and moreover

Let $\xi_{1}, \ldots , \xi_{n}$ be random variables having finite expectations. Prove that

and

Give an example of random variables $\xi$ and $\eta$ such that $\mathbb {E}\left[\xi \right]$ and $\mathbb {E}\left[\eta \right]$ exist, but $\mathbb {E}\left[\xi \eta \right]$ does not exist.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of nonnegative random variables. Prove that

if the series $\sum_{n = 1}^{\infty } \xi_{n}$ converges with probability 1.

Let $\eta , \xi , \xi_{1}, \xi_{2}, \ldots$ be random variables on ($\Omega , \mathscr {A}, \mathbb {P}$) such that at every $\omega$, $\xi_{n} \rightarrow \xi$ as $n \rightarrow \infty$. It is known that $\mathbb {E}\left[\xi_{n}\right]$ need not converge to $\mathbb {E}\left[\xi \right]$ (even if $\mathbb {E}\left[\xi \right]$ exists). Prove that $\mathbb {E}\left[\xi_{n}\right] \rightarrow \mathbb {E}\left[\xi \right]$ if one of the following conditions holds:

Let $A_{1}, A_{2}, \ldots$ be a sequence of independent events. Prove that

Let $A_{1}, A_{2}, \ldots$ be a sequence of independent events. Prove that for the series

to converge, it is necessary and sufficient that

Can the independence condition be replaced by the condition of pairwise independence of the events $A_{1}, A_{2}, \ldots$?

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space such that for any event $A$ the set of all events independent of $A$ forms an algebra. Prove that in this case, pairwise independence of any collection of events implies their independence.

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space. Prove that for all events on it to be pairwise independent, it is necessary and sufficient that every event have probability equal to 0 or 1.

Is the relation of independence on the set of events of an arbitrary probability space transitive?

Is the relation of dependence on the set of events of an arbitrary probability space transitive?

Prove that for the relation of independence on some probability space to be transitive, it is necessary and sufficient that every event of this probability space have probability 0 or 1.

Events $A$ and $B$ are called $\varepsilon$-independent if

Prove that if $A$ is $\varepsilon$-independent of itself, then either $\mathbb {P}\left(A\right) \leqslant 2 \varepsilon$, or $\mathbb {P}\left(A\right) \geqslant 1-2 \varepsilon$.

Let event $A$ be such that $\mathbb {P}\left(A\right) \leqslant \varepsilon$ or $\mathbb {P}\left(A\right) \geqslant 1-\varepsilon$. Prove that $A$ and any event $B$ are $\varepsilon$-independent.

Let $A$ and $B$ be $\varepsilon$-independent events. Prove that the events:

Let $\mathscr {A}_{1}$ and $\mathscr {A}_{2}$ be independent semialgebras. Prove that the algebras they generate are also independent.

Let $\mathscr {A}_{1}$ and $\mathscr {A}_{2}$ be independent algebras. Prove that the $\sigma$-algebras they generate are also independent.

Let ($\Omega , \mathscr {A}, P$) be a probability space, and $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ two sub-$\sigma$-algebras of $\mathscr {A}$, with $\mathscr {B}_{1} \subset \mathscr {B}_{2}$. Can $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ be independent if $\mathscr {B}_{1}$ contains an event whose probability differs from 0 and 1?

Prove that the union of two independent $\sigma$-algebras, each of which contains an event whose probability differs from 0 and 1, cannot be an algebra.

Let $\xi$ and $\eta$ be random variables. Must they be independent if the random variables $\xi^{2}$ and $\eta^{2}$ are independent?

Let $\xi , \eta ,$ and $\zeta$ be random variables, with $\xi$ independent of $\eta +\xi$. Is it true that $\xi$ is independent of $\eta$ and of $\zeta$?

Prove that a random variable is independent of itself if and only if it equals a constant with probability 1.

What conditions must be imposed on $\xi$ for the random variables $\xi$ and $\sin \xi$ to be independent?

Let random variables $\xi$ and $\eta$ be defined on the probability space ($\Omega , \mathscr {A}, \mathbb {P}$), which is the segment $[0,1]$ with the $\sigma$-algebra of Borel subsets and Lebesgue measure. Will $\xi$ and $\eta$ be independent if:

Let ($\Omega , \mathscr {A}, P$) be a probability space, where $\Omega$ consists of exactly $n$ points, each of which has positive probability. Prove that on this probability space there do not exist two independent random variables each of which takes $n$ distinct values.

Let ($\Omega , \mathscr {A}, \mathbb {P}$) be a probability space such that $\mathscr {A}$ contains exactly $n$ events, and let $\xi_{1}, \ldots , \xi_{n-1}$ be pairwise independent random variables defined on this probability space. Prove that at least one of them is a constant with probability one.

Define a probability space ($\Omega , \mathscr {A}, P$) as follows: $\Omega = \left\{ 1,2,3\right\} , \mathscr {A}$ is the set of all subsets of $\Omega$, $\mathbb {P}\left(1\right) = \mathbb {P}\left(2\right) = \mathbb {P}\left(3\right) = 1 / 3$. Prove that on this probability space one cannot define two independent random variables each of which takes at least two values.

A probability space ($\Omega , \mathscr {A}, P$) is defined as follows: $\Omega = \left\{ 1,2,3,4\right\} , \mathscr {A}$ is the set of all subsets of $\Omega , \mathbb {P}\left(1\right) = \mathbb {P}\left(2\right) = \mathbb {P}\left(3\right) = \mathbb {P}\left(4\right) = 1 / 4$. Construct on this probability space two independent random variables not equal to constants with probability one.

Does there exist, on the probability space ($\Omega , \mathscr {A}, \mathbb {P}$) which is the segment $[0,1]$ with the $\sigma$-algebra of Borel sets and Lebesgue measure, a random variable not equal to a constant with probability one and independent of the random variable $\xi (\omega ) = \omega$?

Let $\xi$ and $\eta$ be independent random variables. Prove that the random variables $\min \left\{ 1, \xi \right\}$ and $\min \left\{ 1, \eta \right\}$ are independent.

Let $\xi$ and $\eta$ be random variables such that for any real $a$ and $b$ the random variables $\min \left\{ a, \xi \right\}$ and $\min \left\{ b, \eta \right\}$ are independent. Prove that $\xi$ and $\eta$ are independent.

Let $\xi$ and $\eta$ be independent random variables, and $f(x)$ and $g(x)$ Borel functions. Prove that the random variables $f(\xi )$ and $g(\eta )$ are independent.

Let $\xi$ and $\eta$ be dependent random variables, and $f(x)$ and $g(x)$ Borel functions. Can the random variables $f(\xi )$ and $g(\eta )$ be independent? Does the answer change if we additionally assume that $f(\xi )$ and $g(\eta )$ have nondegenerate distributions?

Can there exist a random variable $\xi$ and two Borel functions $f(x)$ and $g(x)$ such that the random variables $f(\xi )$ and $g(\xi )$ are independent and have nondegenerate distributions? Does the answer change if we additionally assume that $f$ and $g$ are strictly monotonic functions?

Let $\xi$ and $\eta$ be random variables, $\mathbb {P}\left(\xi > 0\right) = \mathbb {P}\left(\eta > 0\right) = 3 / 4, \mathbb {P}\left(\xi +\eta > 0\right) = 1 / 2$. Prove that $\xi$ and $\eta$ are dependent.

Let $\xi$ and $\eta$ be independent identically distributed random variables, $p = \mathbb {P}\left(\xi > 0\right), q = 1-p$. Prove that $p^{2} \leqslant \mathbb {P}\left(\xi +\eta > 0\right) \leqslant 1-q^{2}$.

Let $\xi , \eta ,$ and $\zeta$ be random variables, with $\xi$ independent of $\eta$ and of $\zeta$. Is it true that $\xi$ is independent of $\eta +\zeta$?

Let $\xi , \eta ,$ and $\zeta$ be independent as a collection random variables. Will the random variables $\xi$ and $\eta +\zeta$ be independent? Does the answer change if we assume only pairwise independence of $\xi , \eta ,$ and $\zeta$?

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables, and $\varphi \left(x_{1}, \ldots , x_{k}\right)$ and $\psi \left(x_{1}, \ldots , x_{n-k}\right)$ Borel functions of $k$ and $n-k$ arguments, respectively. Prove that the random variables $\varphi \left(\xi_{1}, \ldots , \xi_{k}\right)$ and $\psi \left(\xi_{k+1}, \ldots , \xi_{n}\right)$ are independent.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables defined on some probability space ($\Omega , \mathscr {A}, \mathbb {P}$). Prove that on this same probability space there exists a double sequence of random variables

satisfying the following conditions:

1. all $\xi_{ij}$ take at most countably many values;

2. for each fixed $k$, $\xi_{1k}, \xi_{2k}, \ldots$ are independent;

3. for each fixed $n$, the sequence $\xi_{nk}$ converges, uniformly in $\omega$, to $\xi_{n}$ as $k \rightarrow \infty$.

Do there exist random variables $\xi$ and $\eta$, not equal to constants with probability one, such that:

Show that the equality $\mathbb {E}\left[\xi \eta \right] = \mathbb {E}\left[\xi \right] \mathbb {E}\left[\eta \right]$ does not, in general, imply the independence of $\xi$ and $\eta$.

Prove that if the random variables $\xi$ and $\eta$ take two values each, then the equality $\mathbb {E}\left[\xi \eta \right] = \mathbb {E}\left[\xi \right] \mathbb {E}\left[\eta \right]$ implies the independence of $\xi$ and $\eta$.

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables with finite expectations. Prove that

Let $\xi$ and $\eta$ be independent random variables, and $\varphi (x)$ and $\psi (x)$ Borel functions. Prove that