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2 votes
0 answers
86 views

Bounds on number of distinct substrings

I have a table with $r$ rows of length $\ell$, with each cell containing a letter from an alphabet $A$ of length $a$. I'm trying to determine the expected number of distinct strings of length $k$ ...
akuan30000's user avatar
2 votes
0 answers
419 views

Best possible concentration inequality in high dimensions

Let $X_1,\ldots,X_n$ be independent random variables in $\mathbb{R}^d$ with $EX_i=0$ and $||X_i||_{2}\leq 1$. What is the best known exponential upper bound for $$P(||X_1+\cdots+X_n||_{2}>x)?$$ In ...
TOM's user avatar
  • 2,288
2 votes
0 answers
83 views

Asymptotic results in unbalanced left $d$-regular expander graphs

Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$. Call a graph $G = (U, ...
rodms's user avatar
  • 409
2 votes
0 answers
143 views

Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function $\mathcal{H}...
Alex R.'s user avatar
  • 4,952
2 votes
0 answers
186 views

Finding the number of leaf nodes at specific level of a random tree

Given a uniform recursive tree (URT) of size $N$ rooted at one node whereby the tree is generated as follows: Starting with a root node, at each iteration, a new node is connected to one of the ...
Val K's user avatar
  • 355
2 votes
0 answers
979 views

How to calculate/approximate expectation of function of a binomial random variable?

Hi, I am stuck at following problem in my research. Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\...
Navneet M's user avatar
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 ...
Yaroslav Bulatov's user avatar
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 ...
Eduardo Lopez's user avatar
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 $...
James Propp's user avatar
  • 19.7k
1 vote
2 answers
306 views

Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
1 vote
2 answers
163 views

Coupling a binomial - parity conditioning

If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (...
DJA's user avatar
  • 435
1 vote
2 answers
535 views

Randomized algorithm?

The problem is as follows. Given a set $S$ of natural numbers of size $n$ where each $x_i \in S$ is from the set $[n^2]$. Elements of $S$ are not necessarily pairwise different, i.e., there can be ...
Joe's user avatar
  • 113
1 vote
2 answers
242 views

Number of required trials to sample all possible states of a 'd'-sided loaded die

Let's say that I have a loaded $d$-sided die where the relative probabilities for the die landing on a particular side, $(p_1, ..., p_d)$, are known. How many times must I roll the die to, on average,...
user14324's user avatar
  • 309
1 vote
1 answer
170 views

Mean of probability distribution

I have a probability distribution defined by the following density function: $f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
Cardstdani's user avatar
1 vote
2 answers
2k views

Proving a random bipartite graph contains a perfect matching

I have the following problem consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
Pavan Sangha's user avatar
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 ...
vonjd's user avatar
  • 5,935
1 vote
1 answer
318 views

How to calculate this limit (if exist)?

I have just asked the calculation of the following summation see here $$S(a,b,m,n_1,n_2)=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}, $$ which is motivated by the calculation of the ...
Dian's user avatar
  • 57
1 vote
2 answers
302 views

Counting permutations defined by a simple process

Consider $n$ labeled balls, $k$ of which are red and $(n-k)$ blue. Given a permutation of these balls, we tick $n-1$ times. For the $i$-th tick, if the $i$-th ball in the permutation is red, then it ...
macat's user avatar
  • 155
1 vote
1 answer
141 views

Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$

This is based on an older question. For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $...
Dominic van der Zypen's user avatar
1 vote
1 answer
154 views

Is the Krawtchouk ensemble a determinantal process?

The Krawtchouk ensemble is defined by a weight: $w(x) = \binom{K}{x}p^x q^{K-x} $ and in fact it comes from a conditioned random walk on $\mathbb{Z}^N$. It is a probability measure on the set $\{ 0, ...
john mangual's user avatar
  • 22.8k
1 vote
1 answer
173 views

Probability of paths to the boundary of a tree

Let $G_n$ be the $4$-regular tree of depth $n$, that is to say the finite graph given by the ball of radius $n$ in the Cayley graph of the free group on two generators. By the root I mean the vertex ...
burtonpeterj's user avatar
  • 1,769
1 vote
1 answer
363 views

limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit: $\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$ When $...
Bruno Brogni Uggioni's user avatar
1 vote
1 answer
1k views

Hamming distance distribution induced by binary hypercube

The following problem arises in a particular machine learning problem: Assume that we have $n$ independent Bernoulli random variables with parameters $p_i$, e.g. $n=5$ and the $p$ vector is $(0.2, 0....
Stephan D.'s user avatar
1 vote
1 answer
1k views

Average Hamming distance between strings after some number of random substitutions in a population of initially identical elements

Let's say I have a set $S$, $(s_1, ..., s_i, ..., s_P) \in S$, of $P$ identical strings over a $k$-letter alphabet, each of length $|s_i| = L$. With uniform random probability across all strings in $...
user14324's user avatar
  • 309
1 vote
1 answer
187 views

Expected cardinal of the intersection between a random subset and a fixed subset

I have a set of size $n$, and a fixed subset $A$ of cardinal $k$. I take a random subset $X$ of cardinal $d$. I need to compute the expected cardinal of the intersection between $A$ and $X$. I tried ...
gthev's user avatar
  • 11
1 vote
1 answer
268 views

Entropy upper bound for the union of uniform distributions over union-closed families

The following question is motivated by the recent breakthrough result by Justin Gilmer on the union-closed sets (aka Frankl) conjecture. Let $\mathcal{F}\subseteq\mathcal{P}(\mathbb{N})$ be a finite, ...
RaffaeleScandone's user avatar
1 vote
1 answer
173 views

Could you provide some TSP examples from real world to test a new algorithm?

It's well known that to find a hamilton cycle is NPC, while TSP is NPH. But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ...
shen lixing's user avatar
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 $v$ ...
D_809's user avatar
  • 175
1 vote
1 answer
638 views

Threshold for perfect Matchings in Bipartite graph

Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$. I know one way to prove the threshold of a perfect matching is ...
Pavan Sangha's user avatar
1 vote
1 answer
269 views

A problem in symbolic dynamics

I got a fun problem. Define the alphabet $\mathcal{A}=\{0,1,2\}$ and the set $\mathcal{A}^{\leq n}=\{ x_1x_2\ldots x_n: x_i\in \mathcal{A}\}$ of words of length $n,$ for each $n\in\mathbb{N}.$ ...
user39115's user avatar
  • 1,805
1 vote
2 answers
399 views

Sums Of Independent Random Variables: Pathological Behaviour

Background: The result of a chess game between two players is a win ,a loss or a draw which are (usually) scored respectively $1$ point, $0$ point or $0.5$ point for the appropriate player. Team ...
Ian Calvert's user avatar
1 vote
2 answers
276 views

What is the probability for sequence of lenght L in subset of [n]

I am trying to calculate the probability that i'll have L length sequence in a random subset of [n] when the subset size is k. for example, if n=5, k=4 and L=2 I'll have the below subsets: {2,3,4,5}, {...
Pedro's user avatar
  • 11
1 vote
3 answers
501 views

Operator probability in a RPN string

Consider the set $S_n$ of all strings of length $n$ ($n$ integer, $n \geq 3$) representing an expression in RPN ( http://en.wikipedia.org/wiki/Reverse_Polish_notation. ) Assumptions (to simplify): ...
Luna's user avatar
  • 31
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 ...
Paul Reiners's user avatar
1 vote
1 answer
107 views

Iterated optimal transport

Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
tex.support's user avatar
1 vote
1 answer
75 views

Probability of correctly guessing the maximum event probability of a multinomial distribution

I have a sample from multinomial distribution with $n$ trials, and $k=3$ options. I know that one of the event probabilities $p_i$ is larger than the two others (who are equal). I'm trying to guess ...
Ted's user avatar
  • 267
1 vote
1 answer
119 views

Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?

We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
Bernard Vatant's user avatar
1 vote
1 answer
207 views

Expectation of the sum of the squares of the cardinal of an inverse function

I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as: $$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$ where $\oplus$ is the bitwise XOR. ...
Tristan Nemoz's user avatar
1 vote
1 answer
106 views

Almost-parallel corners of the hypercube in high dimensions

Say I would like a collection of k "almost-parallel" boolean vectors $X_1,...,X_k \in \{\pm 1\}^n$, such that $(X_i,X_j)/n \approx 1-\epsilon$ for some small $\epsilon$. How many ways are ...
DJA's user avatar
  • 435
1 vote
3 answers
339 views

Coupon collector targeting a collection among many

I am interested in the following problem: We are given a universe $U$ of $n$ coupons, partitioned into $k$ collections, $C_1,\dots C_k$. At each time step $t$, a coupon $X_t$ is selected uniformly at ...
GBathie's user avatar
  • 111
1 vote
1 answer
338 views

Polynomial form/Fourier transform of rational function over finite affine space

I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem. Consider the space of sequences of $n$ zero-one ...
Vilhelm Agdur's user avatar
1 vote
1 answer
265 views

Probability a near universal hash function $ax \bmod p \bmod m$ produces an output from inputs equal modulo $m$

For one of the near universal hash functions $f(x) = ax \bmod p \bmod m$ where $p$ is prime and $m < p, m>1$ and $x$ ranges over $1 \dots p-1$ , what is the probability that given $x_r \in \{ x |...
botsina's user avatar
  • 23
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 ...
TOM's user avatar
  • 2,288
1 vote
2 answers
116 views

How to use probability to find a matching in a family of graphs?

In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
Henry Zagreb's user avatar
1 vote
1 answer
134 views

Random optimization problem

Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...
Penelope Benenati's user avatar
1 vote
1 answer
72 views

Independent identical distribution sequence

given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $. I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
jason's user avatar
  • 553
1 vote
1 answer
976 views

Concentration of the load of the maximally loaded bin ($m$ balls $n$ bins) with nonuniform bin probabilities

There is a common argument used when investigating the concentration of the maximally loaded bin (say $X$ is the maximum load) when $m$ balls are thrown into $n$ bins under the uniform distribution. I ...
kodlu's user avatar
  • 10.4k
1 vote
1 answer
114 views

Probability for a group of stones to live on an infinite Go board

Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 ...
Fan Zheng's user avatar
  • 5,169
1 vote
1 answer
215 views

Random walks cover time in random regular graphs

Let $G=(V,E)$ be a random $r$-regular graph on $n$ nodes. Perform a random walk on $G$, starting from a node chosen according to the walk stationary distribution (i.e. chosen uar from $V$). Claim. If ...
emme's user avatar
  • 23
1 vote
1 answer
95 views

Reference for a local density theorem for binary vectors

I have the following theorem written on my whiteboard, but have misplaced the reference. I believe the probabilistic method may be involved in the proof. Any pointers appreciated. Theorem Let $v\in\{...
kodlu's user avatar
  • 10.4k

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