All Questions
70 questions
3
votes
0
answers
303
views
Exchangeable or iid random variables and linear conditioning
Let $X_1,\ldots ,X_N$ be independent identically distributed random variables (or, more generally, exchangeable random variables,
but let's assume independence for simplicity). Then
$$
E(X_i\mid X_1+\...
- 1,765
2
votes
1
answer
150
views
Probability of collision of some family of hash functions
Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
- 321
2
votes
1
answer
272
views
A generalization of negative binomial distribution
Assume we have a set of $n$ balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of ...
- 61
4
votes
2
answers
314
views
Convexity of truncated expectation
Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
- 41
7
votes
1
answer
342
views
Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution
Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...
- 183
3
votes
0
answers
157
views
Growth of inner products between two random vectors on the sparse hypercube
We define the $s$-sparse hypercube in $\mathbb{R}^d$ as
\begin{align}
\mathbb{H}_s = \bigl \{ {\bf{v}} \in \{ -1, 0 , 1\}^d \colon \| {\bf{v}} \|_0 = s \bigr\},
\end{align}
where $ \| {\bf v} \|_0 $ ...
- 1,127
6
votes
0
answers
277
views
universality for large deviations?
This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
- 3,587
6
votes
2
answers
2k
views
Distribution of $\max_{n \ge 0} S_n$, random walk
Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...
- 61
1
vote
0
answers
255
views
Multiple Bipartite graphs and matchings
I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
- 903
1
vote
0
answers
273
views
A natural sum over multisets (expectation over multinomial)
I think this is a natural question but am not sure where to find resources.
Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...
- 4,529
0
votes
0
answers
216
views
Computation on Random Bipartite graphs
I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
- 903
8
votes
1
answer
227
views
Distribution of entries of a doubly-sorted random matrix
Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...
- 2,479
4
votes
2
answers
380
views
Joint probability distribution as functions
Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal B}...
- 305
12
votes
3
answers
2k
views
How to efficiently sample uniformly from the set of $p$-partitions of an $n$-set?
Let $n,p \in \mathbb{N}_+$ with $p \leq n.$ Let $\mathcal{P}$ denote the set of partitions of $\{1, \ldots, n\}$ into $p$ nonempty sets. How can I efficiently sample uniformly from $\mathcal{P}$?
- 922
14
votes
3
answers
2k
views
Concentration bounds for sums of random variables of permutations
I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
- 539
2
votes
1
answer
635
views
Azuma's Inequality when the conditions hold with high probability?
In Azuma's Inequality, is the statement true when $|X_k - X_{k-1}| < c_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the ...
- 201
3
votes
1
answer
376
views
The degrees in a random subgraph
Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$.
Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
8
votes
3
answers
789
views
A Variance-Tail Description for Continuous Probability Distributions
Start with a continuous probability distribution given by a density function f(x). Let X be a real random variable whose distribution is given by the probability distribution.
I would like to ask ...
- 24.7k
4
votes
2
answers
295
views
Distribution of the biggest gap
Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a_1,a_2,\dots,a_n\)}$, and then we can ...
- 177
21
votes
1
answer
32k
views
How to compute KL-divergence when PMF contains 0s?
From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...
- 439