Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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146
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7answers
9k views

Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
50
votes
7answers
8k views

Chance of meeting in a bar

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly ...
40
votes
8answers
56k views

What is the best book to learn probability?

Question is quite straight... I'm not very good in this subject but need to understand at a good level.
39
votes
10answers
5k views

The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
38
votes
5answers
5k views

Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, ...
37
votes
7answers
9k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
34
votes
3answers
3k 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 ...
33
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
32
votes
9answers
7k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
29
votes
4answers
31k views

Probability density function vs. probability mass function

I've an confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
28
votes
3answers
2k 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 ...
27
votes
12answers
6k views

Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...
26
votes
3answers
624 views

Expected length of the shortest polygonal path connecting random points

$N$ points are selected in a uniformly distributed random way in a disk of a unit radius. Let $L(N)$ denote the expected length of the shortest polygonal path that visits each of the points at least ...
23
votes
1answer
4k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
22
votes
6answers
9k views

Zero probability and impossibility

I read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never ...
22
votes
1answer
929 views

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far ...
22
votes
2answers
617 views

Does Brownian motion visit every point uncountably many times?

Let $B_t$ be a one-dimensional standard Brownian motion. Is it true that, almost surely, for every $x \in \mathbb{R}$ the set $\{t : B_t = x\}$ is uncountable? Let $A_x$ be the event that $\{t : ...
22
votes
1answer
892 views

Expository articles on Analysis and Probability theory

When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
21
votes
5answers
3k views

What is a Markov Chain?

What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.
19
votes
1answer
750 views

Slowest frog on a ladder amongst many, how fast does it climb and how much is it lagging below the others?

In English: Frogs are climbing up a ladder. Each frog jumps to the next level of the ladder at unit rate and independently of the other frogs and of the level it is at. All the frogs start at level ...
19
votes
2answers
830 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 ...
19
votes
1answer
450 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemple. How to calculate the number of circuits that visit ...
18
votes
2answers
786 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 ...
18
votes
2answers
1k 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 ...
18
votes
3answers
6k views

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
18
votes
3answers
2k views

Card doubling paradox

Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after ...
17
votes
2answers
3k views

Beta function derivation

How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments? I'm pretty sure this is ...
17
votes
7answers
5k views

Good books on “advanced” probabilities

what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of ...
17
votes
1answer
275 views

Show two random variables have same distribution

Let X, Y be two non-negative random variables satisfying the condition $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. How can one show that X and Y are equal in ...
16
votes
3answers
395 views

Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
15
votes
1answer
580 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
15
votes
1answer
262 views

What is the distribution of this random series?

Let $\xi_n$ be iid and uniformly distributed on the three numbers $\{-1,0,1\}$. Set $$X = \sum_{n=1}^\infty \frac{\xi_n}{2^n}.$$ It is clear that the sum converges (surely) and the limit has $-1 \le ...
14
votes
2answers
1k views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
14
votes
1answer
151 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
14
votes
2answers
600 views

Central Limit Theorem implies Law of Large Numbers?

Let $X_i$ be iid random variables and let $\overline{X}_n=(X_1+\cdots+X_n)/n$. If $EX_i=\mu$ and $\operatorname{Var}X_i = \sigma^2$ then the central limit theorem says that with some conditions we ...
14
votes
1answer
273 views

Probability of a zero product given one previous zero product

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each element of $v$ is independently $\pm1$ with prob $1/2$. Each element of $w$ is independently $1$ with probability ...
14
votes
1answer
2k views

What is meant by a continuous-time white noise process?

What is meant by a continuous-time white noise process? In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time ...
13
votes
3answers
4k views

How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$? Where, $$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq ...
13
votes
3answers
6k views

Are functions of independent variables also independent?

It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two ...
13
votes
1answer
3k views

Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
13
votes
3answers
2k views

Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?

Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ ...
13
votes
2answers
867 views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
13
votes
3answers
3k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
13
votes
2answers
434 views

Difference in probability distributions from two different kernels

I wonder if the probability kernels of Markov processes on the same state space are close enough, does it also hold for the probabilities of the event that depend only on first $n$ values of the ...
13
votes
4answers
632 views

Playing the St. Petersburg Lottery until I lose everything

This question continues the following question: Calculating the probability of winning at least $128$ dollars in a lottery St. Petersburg Paradox Here is a lottery: A fair coin is flipped repeatedly ...
13
votes
2answers
329 views

Number of digits until a prime is reached

Begin with a random digit from $1$ to $9$. Add a random digit to the right-hand side from $0$ to $9$ until a prime number is reached. How many digits are necessary in the avarage ? More precisely: ...
13
votes
3answers
392 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
12
votes
2answers
372 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
12
votes
2answers
3k views

Does convergence in distribution implies convergence of expectation?

If we have a sequence of random variables $X_1,X_2,\ldots,X_n$ converges in distribution to $X$, i.e. $X_n \rightarrow_d X$, then is $$ \lim_{n \to \infty} E(X_n) = E(X) $$ correct? I know that ...
12
votes
4answers
7k views

Example where union of increasing sigma algebras is not a sigma algebra

If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra? It seems closed under complement since for all ...