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-...

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180
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8answers
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 ...
59
votes
6answers
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 ...
52
votes
8answers
81k 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.
48
votes
10answers
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 ...
47
votes
5answers
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, ...
44
votes
7answers
12k 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 ...
43
votes
8answers
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 ...
40
votes
4answers
54k 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 ...
36
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 ...
36
votes
3answers
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 ...
34
votes
13answers
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 ...
34
votes
1answer
6k 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, ...
29
votes
6answers
17k 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 ...
29
votes
3answers
741 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 ...
29
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 1....
27
votes
3answers
9k 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 }{\...
26
votes
5answers
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.
22
votes
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 "...
22
votes
1answer
818 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 ...
22
votes
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 ...
22
votes
2answers
801 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 $n$. ...
22
votes
2answers
692 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
979 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
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 ...
21
votes
2answers
575 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<...
21
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 ...
20
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 ...
20
votes
1answer
532 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 ...
20
votes
2answers
964 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 ...
20
votes
3answers
602 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 ...
20
votes
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-...
20
votes
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 ...
19
votes
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 ...
18
votes
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{...
18
votes
2answers
827 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\...
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
188 views

How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how ...
17
votes
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\...
17
votes
3answers
418 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 ...
16
votes
3answers
8k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
16
votes
1answer
17k views

conditional and joint probability (3 variables)

I'm having troubles verifying why the following is correct. $$p(x,y|z)= p(x|y,z)p(y|z)$$ I tried grouping the (x,y) together and split by the conditional which gives me $$p(x,y|z)=p(z|x,y)p(x,y)/p(...
16
votes
2answers
975 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (...
16
votes
1answer
2k views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel $\sigma$-...
16
votes
1answer
277 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 ...
16
votes
1answer
2k views

Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
16
votes
0answers
472 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
16
votes
2answers
358 views

Probability of picking an odd number from the set of naturals?

I know there's no uniform distribution for a countably infinite set, but I'm wondering if there's still a way to determine the probability of picking from a subset of a countably infinite set. For ...
15
votes
3answers
9k 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 $...
15
votes
2answers
5k 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 ...
15
votes
3answers
8k 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 ...