Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
Questions tagged [pr.probability]
8641 questions
56
votes
12 answers
Is pi a good random number generator?
Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, which makes me wonder whether I should just use…
James Propp
- 19,363
40
votes
1 answer
The human body's random number generator
I remember learning in microbiology that the human body generates antibodies using a random process so that an enormous variety of antibodies can be produced with a simple genetic code.
Now that I'm trying to learn more about random processes, I…
Brian Rushton
- 3,309
29
votes
5 answers
Random walk: police catching the thief
I posted this problem on stackexchange.com,but haven't get a satifactory answer.
This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$:
Suppose there is a thief at the origin 0 and N policemen at…
zemora
- 291
26
votes
4 answers
A percolation problem
Let's consider the 2-dimensional integer lattice $\mathbb{Z}^2$ for simplicity. In "ordinary" bond percolation, there is a parameter $p \in [0,1]$, and each edge is on with probability $p$. Consider now the following model : all edges are present,…
Peter Hegarty
- 983
24
votes
1 answer
Expected halting time for "The 2^n Game" (aka 2048) -- with random moves
Recently I encountered an online flash game that features an m-by-m grid and input from the directional pad (up, down, left, right). At any point in the game, the grid contains numbers ('blocks') from the set $\{2^i\},1\le i\lt n$. So at each step…
user48413
20
votes
9 answers
Random vs Unknown
Is there any distinction at all between a random quantity and an unknown quantity or is it impossible to distinguish?
Example: 5 minutes in the future, I plan to roll a die the the number of the die roll at that point in time is called X. Suppose…
user1149012
- 197
20
votes
4 answers
Is there a statistical interpretation of Green's theorem, Stokes' theorem, or the divergence theorem?
This is cross-posted from math.stackexchange and stats.stackexchange. Probably there is no great answer to this question, but I thought I'd give it a shot here.
I'm teaching a class on integration of functions of several variables and vector…
Paul Siegel
- 28,772
20
votes
3 answers
what is the probability that a scissor became the champion?
Here is a question from one of my students:
suppose 8 players are in an elimination match. The players are marked with marked with either R (for rock), P (for paper) or S (for scissors). If two players are marked with the same letter, then one is…
user16674
- 201
- 1
- 4
19
votes
0 answers
support of the coupling between two probability measures
Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the marginal distribution of the first and second…
gondolier
- 1,829
19
votes
3 answers
Anti-concentration of Bernoulli sums
Let $a_1,\ldots,a_n$ be real numbers such that $\sum_i a_i^2 =1$ and let $X_1,\ldots,X_n$ be independent, uniformly distributed, Bernoulli $\pm 1$ random variables. Define the random variable
$S:= \sum_i X_i a_i $
Are there absolute constants…
Luca Trevisan
- 363
19
votes
7 answers
A geometric interpretation of independence?
Consider the set of random variables with zero mean and finite second moment. This is a vector space, and $\langle X, Y \rangle = E[XY]$ is a valid inner product on it. Uncorrelated random variables correspond to orthogonal vectors in this space.…
angela
- 415
18
votes
2 answers
What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order?
What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from the distribution of this N-dimensional…
Matthew Lloyd
- 183
- 7
18
votes
3 answers
Existence of a "quasi-uniform" probability distribution on $\mathbb{Z}$
Does there exist a probability distribution on $\mathbb{Z}$ such that
for every integer $n\geq 1$, the probability that a random integer $x$
is divisible by $n$ equals $1/n$?
Henry Cohn has an argument why this is not possible, but it is…
Richard Stanley
- 49,238
17
votes
4 answers
How many dimensions is it safe to get drunk in?
In Michael Lugo's blog post Variations on the drunken-bird theorem, and real-world sightings he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in might be.
So where's the line between "always…
petef
- 1
17
votes
1 answer
Uniformly integrable sequence such that a.s. limit and conditional expectation do not commute
Hi,
Could anyone give an example such that:
$$Y_i \rightarrow Y_{\infty}, \text{a.s.},$$
and $Y_i$'s are uniformly integrable.
But $\mathbb{E}(Y_i|\mathcal{G})$ does not converge a.s. to $\mathbb{E}(Y_{\infty}|\mathcal{G})$ for some sub-$\sigma$…
john KING
- 191
- 3