All Questions
Tagged with pr.probability co.combinatorics
802 questions
9
votes
1
answer
1k
views
Points on binary hemispheres of the n-sphere
Let $\mathbb{S}^{n-1}=${$ x\in \mathbb{R}^n| \sum_{k=1}^n x_k^2 =1 $} be the $n-1$ sphere and $n_i\in\mathbb{R}^n$ with components $n_{ij}\in${$-1,1$}$\ \forall\ j=1,2,\dots,n$. There are obviously $2^...
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- 1
4
votes
3
answers
439
views
Probability estimates for "beans & boxes"
From a discussion with some friends, this apparently easy problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial ...
CommunityBot
- 1
7
votes
2
answers
2k
views
Assigning positive edge weights to a graph so that the weight incident to each vertex is 1.
Let $\Gamma=(G,E)$ be a connected undirected graph, with no loops or multiple edges. $G$ is finite or countably infinite. For each edge $e=\{x,y\}\in E$, we assign a positive, symmetric edge weight $...
- 85.6k
0
votes
1
answer
292
views
Probability of preserving connectivity between pair of vertices in weighted graph
Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.
Let's fix two connected vertices $v_1, v_2 \in V$.
Random graph $G'=(V,E')$ is obtained from $G$ by ...
- 54.2k
24
votes
1
answer
2k
views
A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.
I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?...
- 60.5k
5
votes
1
answer
2k
views
Random points in a rectangular grid defining a closed path
Suppose we have a $n\times m$ rectangular grid (namely: $nm$ points disposed as a matrix with $n$ rows and $m$ columns).
We randomly pick $h$ different points in the grid, where every point is ...
- 6,342
9
votes
1
answer
695
views
Probability of return vs. probability of return in minimal number of steps
Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ ...
CommunityBot
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5
votes
1
answer
394
views
Product of coordinates of a random point from Hamming sphere
Let us consider a boolean hypercube $C = \{-1, 1\}^n$. Let $S = \{x \in C \mid |\{i \mid x_i = -1\}| = \varepsilon n\}$ be a Hamming sphere in $C$ (here $\varepsilon$ stands for the fixed parameter ...
- 109k
6
votes
2
answers
729
views
Has the following kind of (minimum degree $d$) random graph been studied?
The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to ...
- 85.6k
9
votes
1
answer
860
views
Random walk on a simple finite network
Consider a graph $\Delta_N = \lgroup (x,y)\in\mathbb{Z}^2| x+y\leq N-1, x\geq 0,\ y\geq 0 \rgroup$ (set of edges is defined in a natural way): see here ).
Take a random walker that wonders around ...
- 1,411
4
votes
2
answers
1k
views
Balls-and-bins type problem
Suppose I have an n-by-n array of bins, and I want to choose k (k >= n) bins from these n^2 bins such that each row of the array has at least one bin chosen. How many ways are there of doing this?
...
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21
votes
11
answers
4k
views
What are some good examples of non-monotone graph properties?
It seems that many, if not almost all, of the properties studied in graph theory are monotone. (Property means it is invariant under permutation of vertices, and monotone means that the property is ...
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6
votes
2
answers
615
views
Optimally directing switches for a random walk
If you are sometimes called upon directing a random walk in a directed graph, how should you direct it so as to maximize the probability it goes where you want?
Formal statement
More specifically, ...
- 6,342
4
votes
1
answer
2k
views
Square of Binomial Coefficient
Background
I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. ...
- 96.4k
5
votes
1
answer
1k
views
Self Avoiding Walk Enumerations
Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic ...
CommunityBot
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14
votes
3
answers
8k
views
Analog of Chebyshev's inequality for higher moments
I have a positive random variable $X$ with $E[X] = 1$ and a small number $k$ more moments bounded by constants:
$$E[(X-1)^i] = O(1) \forall i \in \{2, ..., k\}.$$
I'd like to bound the average of $n$...
- 5,721
3
votes
1
answer
927
views
How to choose $L$ size-$m$ subsets of $\{1,\ldots,n\}$ to maximize expected max overlap with another randomly chosen subset?
GIVEN: Positive integers $n,m,L$ and probabilities $p_1, p_2, \ldots, p_n$.
GOAL: Choose $L$ size-$m$ subsets $S_1, S_2, \ldots, S_L$ of $\{1,2,\ldots,n\}$ to maximize $\displaystyle \mathbb{E}[ \...
CommunityBot
- 1
9
votes
1
answer
1k
views
A Game of Knights and Queens
Let $m,n,u,v \in \mathbb{N}$ be parameters with $m,n \geq 3$. Suppose two players play a game on a $m \times n$ chess board and we denote the squares of the board by the set of points $ (i,j) $ such ...
CommunityBot
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4
votes
0
answers
580
views
Monotonic properties of harmonic functions on graphs
I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
- 965
11
votes
2
answers
880
views
Covering a random graph with spanning trees.
Let $G=(V,E)$ be a connected graph, say $V=\{1,\ldots,n\}$. Let $F=(V,E')$ be a uniformly random forest in $G$. (In other words, $E'$ is a subset of edges $E$ not containing a cycle, and it is ...
- 173
5
votes
1
answer
1k
views
Concentration of Measure for Power Law
I have a power law distribution $X$ with exponent $c$: $$p(X=t) = \left\\{\begin{array}{cl}(c-1)/t^{c} & t \geq 1 \\\\ 0 & t < 1\end{array}\right.$$
From $X$ I take $n$ independent samples ...
CommunityBot
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0
votes
0
answers
319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
10
votes
2
answers
602
views
What is the probability that every pair of students is at some point in the same classroom?
A cohort in a school consists of 75 students who study for 6 years. Each year, the students are randomly distributed into 3 classrooms of 25 students each. What is the probability that, after 6 years, ...
CommunityBot
- 1
2
votes
2
answers
391
views
linear ordering of color balls
Suppose that $n+m$ balls of which $n$ are red and $m$ are blue, are arranged in a linear order, we know there are $(n+m)!$ possible orderings. If all red balls are alike and all blue ball are alike, ...
- 32.1k
4
votes
2
answers
1k
views
expected values over binomial distributions
In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution:
$$F(n)...
- 3,283
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
- 32.1k
3
votes
1
answer
693
views
Sequence of p draws without replacement with biased probabilities
Hi
I have a problem which i find hard to modelize.
Suppose i have an urn with $N$ marbles. Among these marbles, one is white and all the other ones are black. I draw $P$ marbles without replacement. ...
- 6,711
2
votes
0
answers
351
views
Distribution of transformed multinomial variable?
Suppose we have a uniform multinomial distribution over $2^d$ outcomes. Multinomial coefficients give distribution of vector valued variable $v$ where $v$ is the vector of observed counts.
Is there a ...
- 2,242
2
votes
0
answers
530
views
About generalization of stirling numbers of the second kind
Hello,
The Stirling numbers of the second kind count how many ways can a set of $k$ elements be partitioned into $n$ non-empty classes, with $k=n,n+1,\dots$.
My question is: Is there a ...
6
votes
2
answers
1k
views
diameter of a graph with random edge weights
Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be
an exponential random ...
- 4,922
4
votes
2
answers
853
views
Can you explain the description of the Lovasz Local Lemma by Moser+Tardos?
The Lovász Local Lemma (or LLL) concerns itself with the probability of avoiding a collection of "bad" events A, given that the set of events is "nearly independent" (each bad event A &...
- 5,721
3
votes
1
answer
539
views
Probability of generating the symmetric group
The statement is simple:
What is the probability that a set of n-1 transpositions generates the symmetric group, $S_n$?
The motivation is that I remembered reading that this was an open problem ...
- 17.9k
27
votes
2
answers
812
views
What is the right notion of self-dual (two-dimensional) percolation in R^4?
For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
- 4,014
4
votes
6
answers
751
views
Reconstructing an ordering of a multiset from its consecutive submultisets
We have a multiset $S$ of size $t$ with $r$ distinct elements, where $t$ is much larger than $r$. We want to reconstruct an ordering $s_1, s_2, ... s_t$ of the elements of $S$ given the values of $t$ ...
CommunityBot
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8
votes
1
answer
519
views
devise a joint distribution of $\alpha$ and $\beta$
If we assume probability density distribution functions of random variables $\alpha$, $\beta$ and $\alpha/ \beta$, we would like to devise a joint distribution of $\alpha$ and $\beta$. Although ...
CommunityBot
- 1
3
votes
1
answer
610
views
Looking for a probability distribution
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 ...
- 3,631
13
votes
2
answers
3k
views
The probabilistic method - reference to less challenging questions
I am teaching a course in combinatorics and large part of it is dedicated to the probabilistic method especially in the case of graphs. The course is an undergraduate level (almost none of the ...
- 502
25
votes
3
answers
2k
views
Some models for random graphs that I am curious about
G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...
- 24.7k
4
votes
1
answer
232
views
Negative Association of Component Size in Random Hypergraph
I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a ...
- 151k
4
votes
3
answers
286
views
Medium-Sized Calculations and Organization
This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
- 146
2
votes
1
answer
467
views
Distribution on permutations derived from probability of pairwise orderings
A followup question to Probability estimates for pairwise majority votes - I think it doesn't actually give an answer in any terribly precise sense, but it would give something I'd be happy to use in ...
CommunityBot
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1
vote
3
answers
312
views
Chance of something being fixed [closed]
I'm fixing a software defect that occurs 1 in n test runs. If I want to know that the probability of it being fixed is >= p for some 0 <= p < 1, how many times, m, do I need to run the test ...
- 1
4
votes
1
answer
938
views
Random projection and finite fields
Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of ...
- 4,776
1
vote
0
answers
207
views
understanding some derivation in random XORSAT problem
This question is concerned about the paper "The 3-XORSAT threshold" by O. Dubois, J.Mandler. Here is the link: http://dx.doi.org/10.1016/S1631-073X(02)02563-3
Basically one would like to know when is ...
- 25.8k
3
votes
1
answer
366
views
Random generation of subsets using conditional probabilities
Edit: Rewritten with motivation, and hopefully more clarity.
I'm building a site for a card game called dominion. In it, people build 'decks' of 10 distinct cards from a set of (currently) ...
- 13k
1
vote
1
answer
783
views
Probability of n k-sided dice showing exactly m different faces
I found the following closed form solution for the abovementioned problem:
$${1\over k^n}\cdot{k!\over (k-m)!}\cdot{\{{n\over m}\}}$$ with ${\{{n\over m}\}}$ being the Stirling Number of the second ...
- 28k
2
votes
2
answers
2k
views
Hitting times for an N-dimensional random walk on a lattice with (strictly positive) random integer steps
Please consider a random walk on a finite N-dimensional lattice with vectors $(x_1, ..., x_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away ...
- 599
4
votes
3
answers
579
views
Average distance between numbers of the form $2^{a}3^{b}$
I want to order all numbers of the form $2^a3^b$. I need to find the average distance between a random consecutive pair.
For example, in case of a random consecutive pair $2^{n'}$ and $2^{n'+1}$, the ...
- 126
-1
votes
1
answer
502
views
Name for probabilistic version of Pascal's identity and differentiation formula for binomial distribution
I'm trying to find a standard name or standard reference for two simple-to-prove relations involving binomial distributions.
Define:
$b(n,r,p) := \binom{n}{r}p^r(1 - p)^{n-r}$
i.e., it is the ...
- 14k
7
votes
2
answers
627
views
Probability vertices are adjacent in a polygon
With regard to my original question:
A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
I suppose that the responses ...
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