5
votes
2answers
268 views

Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for. I'm interested to know ...
8
votes
2answers
595 views

Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
0
votes
1answer
203 views

two correlated processes

I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out. Assume that there are two ...
1
vote
1answer
232 views

Sum of covariance matrix of products of dependent variables

Consider the sequences of random variables $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$, as well as the corresponding sequence of products, $\{X_i Y_i\}_{i=1}^n$. All $X_i$ share the same mean value, ...
0
votes
1answer
118 views

Expected rank of players in a Bradley-Terry round-robin tournament

Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda_{i}$, and the probability that player $i$ defeats player $j$ is ...
1
vote
0answers
332 views

Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that \begin{equation} \Sigma_{i=0}^\infty f(i) = 1, \end{equation} where \begin{equation} f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...
3
votes
1answer
175 views

Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...
5
votes
2answers
443 views

Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function. Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
6
votes
2answers
363 views

Most inconsistent ranking

A matrix of $k$ rows and $n$ columns is filled with the numbers $1,2,\ldots,k$ such that the following conditions are satisfied: Every column contain all the numbers form 1 to $k$ without ...
0
votes
2answers
258 views

Creating composite rank [closed]

Problem: Suppose that $K$ different students are ranked based on $N$ different parameters (such as Physics marks, English marks, Biology marks, IQ etc). The rank under each parameter can be repetitive ...
11
votes
2answers
485 views

Covariance of INID order statistics [closed]

In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...
7
votes
4answers
1k views

Recent impressive combinatorial developments in probability theory

In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv) ... I suspect that, for at least a decade, the most ...
2
votes
0answers
125 views

finding rank-3 tensors compatible with a rank-2 tensor projection

I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...
4
votes
1answer
128 views

Mean occurrences of letters in complete strings given by a Bernoulli scheme

Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
3
votes
0answers
269 views

Another generalized coupon collector's problem

Suppose there are $L$ types of coupons, the probabilities that they appear are $a_1,a_2,\ldots,a_L$ respectively, $\sum_i^La_i=1$. Each of them is associated with a constrain number ...
5
votes
1answer
209 views

Is the maximum tree-path length distributed lognormally (in the limit) ?

Consider a full binary tree with $k>10$ levels. Let the lengths of individual edges in this tree be i.i.d. random variables with finite moments. Then total lengths of the $2^{k-1}$ source-to-sink ...
6
votes
2answers
345 views

finding the $n$ closest pairs between $2n$ points

Given $2n$ points $x_1, x_2 \ldots x_{2n}$ and a distance $d_{i,j}$ defined between them, how can I best find the set $P$ of mutually exclusive pairs $(i,j)$ such that the sum of their distances $$ ...
2
votes
0answers
368 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. ...
3
votes
0answers
484 views

A combinatorial bound involving Stirling numbers of the second type

My previous question was solved in a very elegant way, hopefully this (seemingly more complicated) case is also easy for experts. I need the inequality ...
4
votes
1answer
549 views

A bound involving Stirling numbers of the second kind and the asymptotics

Let $S_{n,r}$ denote the Stirling number of the second kind. Define $A_{n,r}:=\frac{\binom{n+r-1}{n}(n+r)!}{S_{n+r,r}r!}$. I want to prove: $A_{n,1}\ge A_{n,2}\ge..\ge A_{n,r}\ge \lim_{r\to\infty} ...
4
votes
3answers
354 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 ...
8
votes
7answers
3k views

Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...
0
votes
0answers
276 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} ...
3
votes
3answers
526 views

Hubbiness of a graph

Is there any statistic that can tell how "hubby" is a graph? By this I mean a number that is small when a graph has no hubs, that is, when all nodes are more or less equal degree-wise, and big when ...
2
votes
0answers
452 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 ...
11
votes
2answers
2k views

Bounding sum of multinomial coefficients by highest entropy one

When does the following hold? $\sum_{(i_1,\ldots,i_k)\in E} \frac{n!}{i_1! \ldots i_k!} \le \exp(n H^*)$ Where $H^*=\max_{(i_1,\ldots,i_k)\in E} -(\frac{i_1}{n}\log \frac{i_1}{n}+\ldots ...
5
votes
3answers
256 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 ...
3
votes
3answers
469 views

association schemes, infinite schemes, semi-schemes, quasi-schemes

Some questions about possibly nonsensical ideas: 1) Can you come up with a definition of an infinite association scheme ? 2) Would infinite association schemes relate to infinite groups the way ...
3
votes
2answers
785 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: ...
3
votes
1answer
298 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) ...
13
votes
2answers
595 views

Archaeogenetics

This question is meant to be applied to recover historic information from genetic data. The following model, is (probably) the simplest possible which takes recombinations into account. First, let ...
3
votes
1answer
329 views

Do all correlation coefficients induce a pseudometric?

The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coefficient. That is: ...
4
votes
0answers
323 views

A Local CLT with large variance

For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
2
votes
1answer
311 views

Parity, Balls and Boxes

Start with a distribution $\mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x 1 through x m ...