# Questions tagged [pr.probability]

Questions in probability theory

200 questions
Filter by
Sorted by
Tagged with
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 ...
4k views

542 views

### Which graph parameters are NOT concentrated on random graphs?

It is well known that many important graph parameters show (strong) concentration on random graphs, at least in some range of the edge probability. Some typical examples are the chromatic number, ...
2k views

477 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 ...
1k views

### What is the expected depth of a randomly generated tree?

I thought about this problem a long time ago, but have no ideas about it. The generating algorithm is as follows. We assume there are $n$ discrete nodes numbered from $0$ to $n - 1$. Then for each $i$...
496 views

### Are monotone Boolean functions in P well-approximated by monotone polynomial-size circuits?

Question 1: Is it true that for every polynomial $p(n)$ and $\epsilon >0$ there is a polynomial $q(n)$ such that every monotone Boolean function on $n$ variables that can be expressed by a Boolean ...
1k views

### 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 ...
577 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 ...
530 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 the balls ...
256 views

1k 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 ...
2k views

### “Almost all objects have property P” vs. “It is easy to test whether an object has property P”

I am interested in any relation between "almost all objects(from a universe) possessing a particular property P" versus "testing whether an object has property P being poly. time decidable". My guess ...
452 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 ...
394 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 ...
352 views

### Is Bayesian updating computationally unfeasible?

Bayesian theory is a very popular theory of probabilities based upon a subjective framework of beliefs. However, subjects and beliefs have to be embodied, meaning to be feasible, it ought to be ...
481 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-...
533 views

### Is there any known CCC closed under a probabilistic powerdomain operation?

Equivalently, is there a known denotational semantics for probabilistic higher-order functional programming languages? Specifically, is there a domain model of pure untyped $\lambda$-calculus extended ...
285 views

### Guessing a low entropy value in multiple attempts

Suppose Alice has a distribution $\mu$ over a finite (but possibly very large) domain, such that the (Shannon) entropy of $\mu$ is upper bounded by an arbitrarily small constant $\varepsilon$. Alice ...
260 views

### Can the “mutual independence” condition in the Lovász local lemma be weakened?

The Lovász local lemma, as stated in Corollary 5.1.2 here, is given as follows. Lemma. Let $A_1, \ldots, A_k$ be events such that each $A_i$ has probability at most $p$ and such that each $A_i$ is ...
675 views

### Statistical distance between uniform and biased coin

Let $U$ be the uniform distribution over $n$ bits, and let $D$ be the distribution over $n$ bits where the bits are independent and each bit is $1$ with probability $1/2-\epsilon$. Is it true that ...
626 views

### High probability events without low probability coordinates

Let $X$ be a random variable taking values in $\Sigma^n$ (for some large alphabet $\Sigma$), which has very high entropy - say, $H(X) \ge (n- \delta)\cdot\log|\Sigma|$ for an arbitrarily small ...
565 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 ...
542 views

### Technical question about random walks

(My original question still has not been answered. I have added further clarifications.) When analyzing random walks (on undirected graphs) by viewing the random walk as a Markov chain, we require ...
I am looking for a theorem which say something like this: if the cover time of a reversible Markov chain is small, then the spectral gap is large. Here the spectral gap means $1-|\lambda_2|$, that is, ...