# Questions tagged [pr.probability]

Questions in probability theory

194 questions
Filter by
Sorted by
Tagged with
971 views

### 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 ...
370 views

### Cheeger's inequality for directed graphs?

Cheeger's inequality can be used to relate the size of the worst cut in the graph to the eigenvalue gap of a simple random walk on that graph. I am wondering if it possible to extend this result to ...
239 views

469 views

### Distributing items randomly into groups of equal size

Given n items (20% type A, 80% type B), I'm looking for a way to distribute them randomly into g groups of equal size. It must be possible for one group to end up with As in the majority but it must ...
156 views

### Adding noise to an assignment

Suppose I'm given a CNF formula with $m$ clauses (and $k$ literals in each clause), with a total of $n$ variables in the formula, where each variable is in at most $c$ clauses, along with a satisfying ...
575 views

### Balls and Bins analysis in the m >> n regime.

It's well known that if you throw n balls into n bins, the most loaded bin is highly likely to have $O(\log n)$ balls in it. In general, one can ask about $m > n$ balls in $n$ bins. A paper from ...
6k views

### 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 ...
462 views

### Constructions better than a random one.

I am interested in examples of constructions in the complexity theory which are better than a random constructions. The only one example of such construction which I know is in the field of error-...
143 views

### Scaling procedures to address false 0's after multiplying probabilities

I need to translate a training algorithm that involves sums and multiplications of probabilities to actual code. For that I need some sort of scaling procedure that allows me to avoid underflows, that ...
296 views

### Beating Nonuniformity by Oracle Access

Informally, we say that a Turing machine $M(\cdot)$ approximates a function $f(\cdot)$ if their outputs on a series of inputs are indistinguishable. More formally, let $L$ be a language, $M(\cdot)$ ...
509 views

### The probability by which Bloom filters preserves the order relation given a similarity measure on sets

A Bloom filter is a data structure for probabilistic set-membership. When adding an item to the set, $k$ bits (whose indices are determined by $k$ different hash functions) are set to 1. To check if ...
429 views

### Borel-Cantelli Lemma and Derandomization

I was reading a paper titled Random Oracles with(out) Programmability. The last paragraph of section 2.3 reads: [Using our novel approach] there is no need to apply well-known classical ...
136 views

### Sampling small separators

Is there a tractable way to define an approximate distribution over small vertex separators of the graph? I'm looking for something along the lines of [Bayati,2008], a way to turn a single "this is a ...
309 views

### Geometric / Visual explanation that the average height of a random binary tree of given size $n$ is asymptotically $2\sqrt{\pi n}$

I just finished reading the proof that the average height of a random binary of given size $n$ is asymptotically $2\sqrt{\pi n}$. I'm now searching for an intuitive, or geometric, or visual proof of ...
650 views

### Can the mutual information of a “cell” be negative?

Please forgive me if this is not the right Stack Exchange (I also posted it at Cross Validated). Please also forgive me for inventing terms. For discrete random variables X and Y, the mutual ...
510 views

### Pairwise independent gaussians

Given $X_1,\ldots,X_k$ (i.i.d. gaussians with mean $0$ and variance $1$), is it possible (how?) to sample (for $m=k^2$) $Y_1, \ldots, Y_m$ such that $Y_i$'s are pairwise independent gaussians with ...
519 views

### Avalanche like stochastic process

Consider the following process: There are $n$ bins arranged from top to bottom. Initially, each bin contains one ball. In every step, we pick a ball $b$ uniformly at random and move all ...
476 views

### 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 ...
536 views

### Chernoff-type inequality for random variable with 3 outcomes

Suppose we have a random variable which takes non-numeric values a,b,c and want to quantify how empirical distribution of $n$ samples of this variable deviates from true distribution. The following ...
273 views

### Expected Halt Time

I'm sorry if the following question seems too obvious. In fact, I oversimplified a much harder problem to this one. Consider the following algorithm, where $0 < p \le 1$ is a constant: ...
1k views

### What is the best way to get a close-to-fair coin toss from identical biased coins?

(Von Neumann gave an algorithm that simulates a fair coin given access to identical biased coins. The algorithm potentially requires an infinite number of coins (although in expectation, finitely many ...
1k views

### What does one mean by heuristic statistical physics arguments?

I have heard that there are heuristic arguments in statistical physics that yield results in probability theory for which rigorous proofs are either unknown or very difficult to arrive at. What is a ...
### Bounds on $E[f(x)]$ in terms of $f(E[x])$ other than Jensen's inequality?
If $f$ is a convex function then Jensen's inequality states that $f(\textbf{E}[x]) \le \textbf{E}[f(x)]$, and mutatis mutandis when $f$ is concave. Clearly in the worst case you cannot upper bound \$\...