Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...
1
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0answers
95 views
When can a measurable mapping be factorized?
Problem 13.3 of Probability and Measure by Billingsley states:
$(\Omega, \mathcal{F})$ and $(\Omega', \mathcal{F}')$ are two
measurable spaces. Suppose that $f: \Omega \rightarrow ...
1
vote
1answer
123 views
Markov processes driven by the noise
Let $\xi_n\in \Xi$ be a sequence of iid random variables with $n \in\mathbb N\cup\{0\}$, which we call a noise process. Construct a process
$$
Z_{n+1} = f(Z_n,\xi_n)\quad(\star)
$$
with $Z_0\in E$ ...
1
vote
0answers
2k views
Expected value of max/min of random variables
I am trying to solve the following problem.
Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
5
votes
1answer
231 views
The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable
I'm trying to prove this conclusion but have some problems with one of the steps.
Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove ...
4
votes
0answers
134 views
Uniqueness of the random variable from its distribution
When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf ...
4
votes
5answers
339 views
Sleeping Mathematician (Sleeping Beauty)
I came across the following thought experiment, and I would like to understand whether the controversy around it is justified.
Imagine an experiment in which a mathematician is put to sleep with some ...
4
votes
0answers
210 views
Inequalities involving the probability density function and variance
I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
3
votes
1answer
227 views
Bus stop probability question
People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform ...
3
votes
1answer
121 views
How expected value is related to density function?
Let $X$ be a random variable on $(\Omega, \Sigma, P)$. The expected value of $X$ is defined as
$$EX = \int X \,dP.$$
But when we calculate $EX$, we often use
$$
EX = \int_{-\infty}^\infty xf(x) dx
...
3
votes
1answer
81 views
Product Measures
Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = ...
3
votes
4answers
135 views
Conjunction fallacy
I was reading this article which has the following question,
Linda is 31 years old, single, outspoken, and very bright. She majored
in philosophy. As a student, she was deeply concerned with ...
3
votes
2answers
294 views
Expectation of a stopping time uniquely determined by a function
Let $(X_t)_{t\ge0}$ be a Markov chain on a finite state space $\Omega$, with transition probability $P$. Let $T$ be a stopping time such that $T=\min \{t\ge 0;X_t \in A \subset \Omega \}$.
If ...
3
votes
1answer
202 views
Feller continuity of the stochastic kernel
Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
3
votes
1answer
137 views
Relation: pairwise and mutually
Suppose we can define a relation $R$
over the sets $X_1, …, X_k$ for any
natural number $k$, note not
specified for a particular $k$. I
was wondering if there is some
definition or conditions ...
3
votes
2answers
135 views
probability -Diverging expectation
As I keep reading probability books, there are always some issues that no one considers.
For example,
for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega ...
2
votes
2answers
105 views
Other way to express $e^{|x|+|y|}$
I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
votes
2answers
113 views
Using Recursion to Solve Coupon Collector
I read a brilliant answer by Mike Spivey on one of the questions and I was wondering how I could use it to solve a coupon collectors problem.
The problem is :
There are coupons labelled 1,2,3...,10 ...
2
votes
2answers
25 views
$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y$
$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y.$
Solution.
$$f_{Y|X}(y|x) = \frac{1}{x^2}, \text{ $x \in (0,1]$, $y \in \mathbb{R}$.}$$
Thus
$$f_{X,Y}(x,y) = ...
2
votes
2answers
88 views
Distribution of sums
I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that ...
2
votes
1answer
221 views
Why does a time-homogeneous Markov process possess the Markov property?
Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
2
votes
5answers
523 views
Some case when the central limit theorem fails
If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite ...
2
votes
2answers
111 views
sub martingales and more
This is a problem on sub-martingales.
Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and
where $S_n$ is a symmetric random walk and $\mu$ is greater than zero.
We ...
2
votes
3answers
272 views
Uniform distribution on $\mathbb Z$ or $\mathbb R$
I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
2
votes
2answers
185 views
Detail in Conditional expectation on more than one random variable
I have $E(X|Y,Z)=0$, $X$ independent of $Y$ and of $Z$ and I want to conclude that $E(X)=0$ ($X,Y,Z$ are real-valued random variables). Okay it seems quite obvious, but if I try to make a strict ...
1
vote
0answers
36 views
What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.
In my text, there is a passage that says:
"Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is:
$$
\begin{align*}
f_{s\mid ...
1
vote
3answers
781 views
Poisson Distribution of sum of two random independent variables $X$, $Y$
$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim ...
1
vote
1answer
225 views
probability question on characteristic function
I got a big problem with my exam practice question on characteristic function. Here are two.
Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
1
vote
1answer
101 views
Continuity of Expected Value
Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$.
Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, ...
1
vote
1answer
325 views
Prove that vector has normal distribution
You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$.
Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$.
You build following random variables, based ...
1
vote
5answers
131 views
Probability distribution functions for the perimeter and space of triangle with fixed radius
Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
1
vote
1answer
322 views
recursive equation for number of white balls
Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls ...
0
votes
1answer
75 views
Expected value of c.d.f when normal distributed
I need help to calculate the expected value of an invertal of a c.d.f function which is normal distributed. I know that
$E(X)=\int^\infty_0 (1-F(x))dx$
What i need is to calculate
$E(w|w \geq ...
0
votes
0answers
52 views
How to model mutual independence in Bayesian Networks?
It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations.
Can mutual ...
0
votes
0answers
138 views
complex integration over the whole plane
I am trying to solve this integral:
$H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$,
where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
0
votes
1answer
144 views
Probability of two opposite events
Suppose there is string of eight bits, e.g.:
00100110
Bits are randomly chosen from the string. All choices are done equally likely.
Probability of choosing $0$:
$p_0 = \frac{5}{8} = 0.625$
...
0
votes
2answers
332 views
Minkowski's Inequality For Infinity
I've tried figuring this out and searching the net on this for 5 hours, but I can't get it. Every source says it's trivial, but I must be missing something because I have pages of work that don't lead ...
-1
votes
1answer
480 views
pdf equation for tossing 2 coins given the probability of landing head for each coin in a single toss
Problem:
Consider a simple coin-flipping experiment in which we are given a pair of coins A and B of
unknown biases, $\theta_{A}$ and $\theta_{B}$ respectively (that is, on any given flip, coin A ...
21
votes
2answers
1k views
Why is this coin-flipping probability problem unsolved?
You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars
the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
23
votes
3answers
835 views
Random Variable Inequality
Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:
(A-S ...
18
votes
2answers
369 views
Cover time chess board (king)
Consider a random walk of a king on a standard chess board, which at each step moves to a uniformly random permitted square. What's the exact mean time to visit all squares (cover time), starting ...
16
votes
2answers
471 views
Translations of Kolmogorov Student Olympiads in Probability Theory
I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward.
I ...
7
votes
2answers
335 views
How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?
I'm learning Kolmogorov's zero-one law in probability theory:
Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in ...
12
votes
2answers
275 views
On the set of the sub-sums of a given series
Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
9
votes
3answers
218 views
Are polynomials dense in Gaussian Sobolev space?
Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of ...
6
votes
1answer
606 views
Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law
I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law?
The former is about limsup of a sequence of events, while the latter is about ...
5
votes
1answer
215 views
For symmetric stable distributions, why is $\alpha \le 2$?
I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact.
Suppose we are trying to come up with stable distributions. From the definition, ...
4
votes
1answer
182 views
Solution to the stochastic differential equation
Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
4
votes
1answer
110 views
Is there one-tailed version of Vysochanskiï–Petunin inequality, like Chebyshev?
The Vysochanskiï–Petunin inequality gives a tighter bound than Chebyshev for unimodal distributions . I'm just wondering if there is a one tailed version of it, like that of Chebyshev inequality? ...
3
votes
1answer
253 views
How to determine the expected value of the $f(x,y)$?
How to determine the expected value of the $f(x,y)$ defined as:
f(x,y):
$\quad$ for i = 1 to y
$\quad$ $\quad$ do x = R(x)
$\quad$ $\quad$ return x
where $R(N)$ returns any ...
3
votes
4answers
856 views
Prerequisites on Probability Theory
Please answer as many questions as you can.
What are the topics one should know before delving into probability theory? (Please recommend any books you know on those topics too.) I think there is set ...
