All Questions
Tagged with co.combinatorics pr.probability
802 questions
8
votes
3
answers
411
views
Identifying a subset with as few tests as possible
Informal description: You are given a set of $n$ blood samples, each having probability $p$ of being infected with a disease. Your goal is to determine the set $P$ of infected samples with as few ...
- 32.5k
9
votes
1
answer
1k
views
Bounding the probability that two binomials are equal
Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.
Let $B_{n,p}$ denote the usual binomial random ...
- 2,720
0
votes
1
answer
78
views
Estimate an expression about probability about Bernoulli random variables
Given $v_{ij} \in \{0,1\}$, $i \in \{1,2\}$, $j \in \{1,2,\ldots,n\}$. Let $X_1, X_2, \ldots, X_n$ be random variables, $P[X_i=1]=P[X_i=0]=1/2$, $i \in \{1,\ldots, n\}$. By checking many examples, I ...
- 6,201
1
vote
0
answers
100
views
Balls and bins, with cardinality constraints
Suppose I have $n$ sets of $k$ balls each, with each one of the $nk$ balls distributed uniformly at random among $m$ bins. Further suppose that I have a probability vector $p=(p_1,\dots,p_m)$. I am ...
- 4,049
3
votes
1
answer
229
views
Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
- 2,720
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
11
votes
1
answer
370
views
Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
- 6,917
8
votes
1
answer
270
views
Sizes of connected components from a random choice in a grid
This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares. ...
- 13.4k
1
vote
2
answers
50
views
Cyclic inequality for 2 dimensional simplex elements
Let $p=(p_{1},p_{2},p_{3})\in\Delta$, with $\Delta:=\lbrace p\in(0,1)^{3}\ |\ p_{1}+p_{2}+p_{3}=1 \rbrace$. I aim to prove (not knowing whether it is true though) that
\begin{equation}
p_{1}^{p_{3}-p_{...
- 289
2
votes
1
answer
195
views
Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S
Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
6
votes
1
answer
527
views
Can information be extracted more precisely using more random trials?
Write $n$ iid draws of $(x,y)$ as $(x^n, y^n)$. Fix $R\in (0,H(x))$. What is the min of $n^{-1}H(y^n|f(x^n))$ over maps $f$ with range $\lbrace 1,\dots,\exp nR\}$, taking $n\to \infty$?
1
vote
0
answers
140
views
Count shortest path with different lengths in random graph
Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are ...
- 1,495
0
votes
1
answer
148
views
A tiling of $\mathbb{Z}^2$ from M. Barlow's paper
In M. Barlow's paper: arxiv.org/pdf/math/0302004.pdf, P17- (2.7) formula.
Let $k\geq 10$, and consider a tiling of $\mathbb{Z}^2$ by disjoint squares
$$T(x):=\{y\in \mathbb{Z}^2: x_i\leq y_i< ...
- 288
3
votes
1
answer
181
views
Infimum of weakly dependent Gaussian process?
Consider some collection of weakly dependent Gaussians $\{w_i\}$ with a uniform bound of $r$ on the magnitude of their covariances. Are there any bounds or techniques towards:
$$E[\inf_i|w_i|] \le f(r)...
- 435
0
votes
1
answer
260
views
Express inclusion-exclusion principle in terms of matrix operations
First of all i denote $\{1,2,3,...,m\}$ by $[m]$
Let there be a collection of sets $\alpha=\{A_{1},A_{2},...,A_{m}\}$ such $\bigcup_{i\in[m]}A_{i}\subseteq [n]$
Consider any function $f:\mathcal{P}([...
- 123
6
votes
2
answers
2k
views
How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?
I have a question about the combinatorial Laplacian $\Delta$ which is defined by
$$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$
where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
- 288
0
votes
1
answer
208
views
Local behavior of the Vandermonde convolution
An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{...
- 5,230
20
votes
3
answers
1k
views
The Angel and Devil problem with a random angel
In the classic version of Conway's Angel and the Devil problem, an angel starts off at the origin of a 2-D lattice and is able to move up to distance $r$ to another lattice point. The devil is able ...
- 6,969
1
vote
0
answers
54
views
Age of the most recent common ancestor for the neutral Wright-Fisher model
The neutral Wright-Fisher model with $n$ individuals is a genealogical model often used in population genetics that can be described as follows: at all generations, there are exactly $n$ individuals, ...
- 11
0
votes
1
answer
158
views
Finding k items in a binary tree
Let us be given a binary tree of height $n$ (and $2^n$ leaves) among which we search $k$ items, where $k < < 2^n$. Suppose we have a test that shows if in the children and childrens-children ...
- 1
6
votes
1
answer
361
views
Random walks on infinite directed regular graphs
Let us consider a directed graph $\Gamma=(V,E,s,t)$ ($V$ set of vertices, $E$ set of edges, $s,t: E \rightarrow V$ are the "source" and "target" maps).
Assume that $\Gamma$ is bi-regular, that is ...
- 26k
8
votes
2
answers
484
views
Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
- 60.5k
1
vote
1
answer
436
views
Size of minimum cut in random graph
Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) The score of each ...
- 1,209
3
votes
0
answers
342
views
Sum of products of irreducible characters of the symmetric group over a subgroup
When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind
$$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \...
0
votes
0
answers
39
views
hypergraph product that preserve expansion properties
I am looking for a hypergraphs product of hypergraph H1,H2 that preserves some expansion properties of H1,H2.
The expansion property I am looking at is HD-random walk.
The product I am looking for is ...
- 340
0
votes
1
answer
101
views
Question of expected number of consecutive coin flip with increasing bias [closed]
This is a question I found on the book and I don't know how to tackle it. Thanks to any help or hint in advance.
I have a coin that, I could get the head 100% at the first flip, $\frac{1}{3}$ at the ...
- 11
1
vote
1
answer
107
views
Concentration of maxima of a random polynomial with Rademacher coefficients
Let $X_1,\ldots, X_n$ be independent Rademacher random variables (i.e. $\mathbb{P}(X_i=\pm 1)=1/2$). Consider the random polynomial $$P_{n}(t)=c+X_{1}t+X_2t^2+\cdots+X_{n}t^n.$$
Is it well known how ...
- 2,288
2
votes
0
answers
80
views
Small set in partition-large class
A collection $\mathcal{A}\subseteq \mathcal{P}(X)$
is $k$-large in $X$
if for every $k$-partition
of $X$ namely
$X_1,\cdots,X_k$, there exists an $i\leq k$ such that $X_i\in \mathcal{A}$;
$\mathcal{...
- 909
1
vote
0
answers
61
views
What is the minimal $m$ for which the independence graph is $n$-universal?
Suppose, an $m$ sided die is rolled. Let's define the independence graph $I_m$ as a graph with the set of all possible events as vertices, and edges between two events iff they are independent.
...
- 2,618
2
votes
1
answer
148
views
Reference request - parallel rectangles discrepancy theory
I've been reading about discrepancy theory and trying to understand some of the open problems in the field. Wikipedia has a list of some of the open problems, but the descriptions are terrible. In ...
- 435
2
votes
2
answers
185
views
Independence depth of linearly dependent random variables
Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise ...
- 2,618
0
votes
1
answer
196
views
Coupling between two distributions
Consider $s = \Theta(n^{\delta})$ for a $\delta\in (0,1)$ and let $p\in (0,1)$ with $m = \lfloor pn\rfloor$. Consider the random variable $Y$ which chooses $m$ elements from $\{1,\ldots,n\}$ such that ...
- 49
1
vote
1
answer
519
views
How typical are integer isometries on a hypercube? Littlewood-Offord problem for Bernoulli Gram matrices
Let $m\geq 3$ be fixed and $n\to\infty$. Consider $v=(v_j)_{j\leq m}$ with $v_1,\ldots,v_m\in \{-1,+1\}^n$. Let:
$N_I(v)$ be the number of sequences $u_1,\ldots,u_m\in \{-1,+1\}^n$ isometric to $v$ ...
- 175
0
votes
1
answer
165
views
Bound for Large deviations of sums of independent (not identical) variables
I am working with a sum of variables $X_i$; they are all independent, but not identically distributed. For any $i$, I can show the bound $$\Lambda^*_{X_i}(t) := \sup_t \langle t, x \rangle - \Lambda_X(...
- 435
0
votes
1
answer
171
views
Closed form solution for a binomial coefficient relation?
In following, $x_{n}$ is a set of given numbers, n = 0, 1, 2, ...,
$y_{n}$ is defined by the following recursive relation of $x_{n}$:
For example:
${\displaystyle {x_{1}=x_{0}y_{1} }}.$
${\...
- 127
7
votes
3
answers
330
views
Quantifying the noninvertibility of a function
Given a function $f$ from a finite set $X$ to itself, it seems natural to consider $\kappa_f := (\sum_{x \in X} |f^{-1}(x)|^2)/|X|$ as a measure of the non-invertibility of $f$: it equals 1 if $f$ is ...
- 19.7k
1
vote
2
answers
203
views
Moments of a combinatorial ensemble of random variables
Suppose $X_1,\dots,X_n$ are jointly distributed random variables such that the random $n$-tuple $(X_1,\dots,X_n)$ is uniformly distributed on the set of $n$-tuples of nonnegative integers summing to $...
- 19.7k
6
votes
2
answers
266
views
Lovasz local lemma for the edge model
In order to successfully apply the Lovasz local lemma, one needs the events to be relatively independent. This (sometimes) works well in the $G(n,p)$ model of random graphs, where the presence or ...
- 2,339
3
votes
1
answer
307
views
Concentration of monochromatic edges in a graph
Let $G$ be a graph of order $n$ with $m$ edges. Color each vertices uniformly at random with $q$ colors. It is easy to see that expected number of monochromatic edges (edge whose end vertices are of ...
0
votes
1
answer
187
views
Proof of consistent of height function
I have a question about the consistent of height function defined on a domino tiling. I always see papers claims that height function is defined consistently. But I am confused with the consistent. ...
user102900
2
votes
0
answers
85
views
How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?
Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...
- 267
12
votes
2
answers
406
views
Does asymmetric fraction of finite groups tend to $0$?
Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
- 2,618
4
votes
1
answer
275
views
Probability of a subset of Bernoulli's being all 1's
Suppose we have $n$ iid Bernoulli's $X_1,\ldots,X_n$ with mean $p$, and a family $\mathcal{F}$ of subsets of $[n]$. The question is how to lower bound the probability that there is a set in the family ...
- 408
2
votes
1
answer
165
views
Covering subset with large probability
Let $c>0$, $0<\lambda<1$, and let $k\in \mathbb{N}$ be sufficiently large. Let $X$ be a uniformly random subset of $\{1,\cdots,N\}$. Denote by $[N]^x$ the collection of $[x]$-element subset ...
- 909
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
1
vote
0
answers
130
views
Probabilistic lower bound on largest singular value of matrices
I have a distribution $\mathcal{D}$ that spits out vectors in $\{-1, 1\}^N$. Suppose I have a sample of $H$ of these vectors which I arrange into a matrix $M$ of the form $H \times N$.
Consider the ...
- 1,129
4
votes
1
answer
272
views
How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?
Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...
- 267
5
votes
1
answer
301
views
Convexity of the expectation of boolean functions
Let $$f:\{-1,1\}^n \to \{-1,1\}$$ be a monotone, odd ($f(-x)=-f(x)$) Boolean function.
Let $$F:[0,1]\to[0,1]$$ denote the probability that $f(x_1,...,x_n)$ where $x_1,...,x_n$ are i.i.d. $\pm1$ R.V. ...
- 61
4
votes
1
answer
115
views
What is the probability of an empty convex $k$-gon among many given points?
Given a finite number of points in the plane in general position, call a convex subset empty if its hull doesn't contain any other of the points.
For a big number $n$ of randomly distributed ...
- 13.4k