# Tagged Questions

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 [tag:probability-...

11k 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 ...
9k 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 ...
79k 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.
7k 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 ...
6k 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, ...
11k 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 ...
10k 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 ...
53k views

### Probability density function vs. probability mass function

I've a 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 ...
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 ...
4k 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 ...
9k 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 ...
5k 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, ...
16k 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 ...
738 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 ...
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 1....
4k 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.
9k views

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 }{\... 8answers 7k 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 "... 1answer 1k 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 ... 2answers 778 views ### Convergence of series \sum\limits_{k=1}^\infty\frac{1}{X_1+\dots+X_k} with (X_k) i.i.d. non integrable Pick a sequence X_1, X_2, \dots, of i.i.d. random variables taking values in positive integers with \mathbb{P}(X_i=n)=\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)} for every positive integer n. ... 2answers 689 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 : ... 1answer 972 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 ... 3answers 2k 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 ... 2answers 568 views ### A simple way to obtain \prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s} Let  p_1 <p_2 <\cdots <p_k <\cdots  the increasing list in set \mathbb{P} of all prime numbers . By sum of infinite geometric series \sum_{k=0}^\infty r^k = \frac{1}{1-r} for 0<... 1answer 814 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 0 ... 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 ... 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 ... 2answers 955 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 ... 3answers 560 views ### Probability and measure theory I'd like to have a correct general understanding of the importance of measure theory in probability theory. For now, it seems like mathematicians work with the notion of probability measure and prove ... 1answer 479 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 ... 1answer 4k 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 ... 1answer 525 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 ... 5answers 27k 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 \mathcal{... 2answers 825 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 I\... 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 ... 2answers 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 wide-sense-... 3answers 5k 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 r\...
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 ...