# Questions tagged [pr.probability]

Questions in probability theory

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### Upper and lower bound of binomial summation

For $n>1$ and $~0<p \leq 1$, can we upper and lower bound the following binomial series in terms of $n$ and $p$ $$\Sigma_{i=\lceil p n \rceil}^n {n \choose i} (p )^i(1-p)^{(n-i)}$$
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### Bounding Rademacher Averages, with and without chaining

One can bound the Rademacher average $R_n(A)$ of a finite set of vectors $A\subseteq\{0,1\}^n$ using Massart's Finite Lemma: $$R_n(A)\le \max_{a\in A}\|a\|\frac{\sqrt{2\ln|A|}}{n}$$ where $\|\cdot\|$...
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### Establishing causality under conditions of certainty

I'm currently reading "Causality: Models, Reasoning, and Inference" by Judea Pearl. Early on, he states that the development assumes that there are no certain entailments, no 1 or 0 probabilities -- ...
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### On a possible probabilistic log-rank conjecture

The communication matrix of a function $f$ is a $2^{n}\times 2^{n}$ matrix $M_f$ where the indices of the rows (columns) correspond to the inputs of Alice (Bob), and each entry $M_f (a,b)$ is the ...
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### Are there efficient general Bonferroni-style bounds known?

A classic problem in probability theory is to express the probability of an event in terms of more specific events. In the simplest case, one can say $P[A \cup B] = P[A] + P[B] - P[A \cap B]$. Let's ...
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### Sampling from Multivariate Gaussian with Graph Laplacian (inverse) Covariance

We know from e.g. Koutis-Miller-Peng (based on work of Spielman & Teng), that we can very quickly solve linear systems $A x = b$ for matrices $A$ that are the graph Laplacian matrix for some ...
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### Strong Dependence

I asked this question on MO, but no answer. I don't know if this definition has been already given. Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$...
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### Throwing Balls into Bins, estimate a lowerbound of its probability

This is not a homework, though it looks like. Any reference is welcome. :-) Scenario: There are $n$ different balls and $n$ different bins (labled from 1 to $n$, from left to right). Each ball is ...
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### Order of drawing colored balls from a bag

We're looking for an upper bound on (or a method to compute) probabilities of the following type: Suppose I put 12 yellow balls, 18 red balls, 7 white balls, and 2 green balls in a bag. Then I start ...
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### Spectrum of absorbing random walk for regular graphs

I have a symmetric Markov chain given by the matrix $P$. Let $M$ be a set of special states of the chain (called marked states, they correspond to solutions to some problem). We can write $P$ as \...
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I have vaguely heard of this connection between random matrix theory and graphs (the spectral gap of their laplacians) on compact Riemann surfaces. Can someone give a pedagogic reference which helps ...
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### Proving properties of Random Graphs

I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas. Consider the random graph model $G_{n,p}$ where its ...
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### What is the probability of a virus spreading through a network given a virus source node?

Model: Consider an infinite undirected connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. At time $t=0$, a given virus node $s\in\mathcal{V}$ starts infecting the network $\mathcal{G}$. ...
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### Independent set size in triangle-free graphs

Consider a triangle-free graph $G$. The notations used are: $\alpha(G) =$ the size of a largest independent set of $G$. $n(G) =$ the number of vertices in $G$. Theorem (Ajtai et al.): For a ...
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### Book on Probability

While I have passed some courses on probability theory, both in the high school and the university, I have a hard time reading TCS papers when it comes to probability. It seems that the authors of ...
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### Does a noisy version of Conway's game of life support universal computation?

Quoting Wikipedia, "[Conway's Game of Life] has the power of a universal Turing machine: that is, anything that can be computed algorithmically can be computed within Conway's Game of Life." Do such ...
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### What is entropy of a variable described by Knightian uncertainty?

Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, as in Dempster-Shafer theory, what is its entropy?
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### The use of crossovers in Genetic Algorithm

My questions concern the use of crossovers in genetic algorithms. The three basic ingredients of genetic algorithms are: selection mutation crossover If we think of genetic algorithm acting on ...
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### Why is the zero value ignored while deriving a ranking function for query terms in Probabilistic IR?

Take for instance equation 67 and 68 from this chapter: the value of $P(q|R=1,q)$ can become zero if the term is not present in the document, and as all probabilities are multiplied, the probability ...
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### question about probability ranking principle

I am studying IR and I am not clearly understand "probability ranking principle" ( I tried to google the definition, but i couldn't find clear answer.) I am assuming that it's system which ...
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### The complexity of sampling (approximately) the Fourier transform of a Boolean function

One thing that quantum computers can do (possibly even with just BPP + log-depth quantum circuits) is to approximate-sample the Fourier transform of a Boolean $\pm 1$-valued function in P. Here and ...
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### Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, i....
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