All Questions
70 questions
1
vote
1
answer
197
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
- 11
1
vote
1
answer
75
views
Probability of correctly guessing the maximum event probability of a multinomial distribution
I have a sample from multinomial distribution with $n$ trials, and $k=3$ options. I know that one of the event probabilities $p_i$ is larger than the two others (who are equal). I'm trying to guess ...
- 267
3
votes
0
answers
81
views
Can we remove the restriction on a parameter in Talagrand concentration inequality?
Recently I am trying to use Talagrand concentration inequality to do something on graphs. I find a version from the book of Molloy and Reed ''Graph Colouring and Probabilistics Method''. I attached a ...
- 1,190
1
vote
0
answers
84
views
How can one build a min-2-wise independent small sample space from min-3-wise permutations?
I have been studying a polynomial-size set of permutations from one of my lectures. The below image, taken from the lecture notes PDF, illustrates how to construct min-3-wise permutations.
My ...
- 43
0
votes
1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
0
votes
2
answers
239
views
Computing the expectation of a quadratic matrix form involving Bernoulli and Gaussian distributed matrices
I am working with two random matrices, $Z$ and $H$:
$Z$ is an $n \times K$ matrix with entries sampled i.i.d. from a Bernoulli distribution: $Z_{ij} \sim \mathrm{Bernoulli}(p)$.
$H$ is a $K \times K$ ...
- 37
2
votes
0
answers
56
views
Dirichlet series solution to Poisson Point Process question (repost from math.SE)
Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
- 121
4
votes
1
answer
197
views
On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
- 128k
3
votes
1
answer
266
views
A linearly distributed version of the balls into bins problem
Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
- 1,677
17
votes
1
answer
1k
views
Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
1
vote
0
answers
663
views
The distribution of hitting time in 2D-lattice random walk [closed]
Assume a particle at $(0,0)$ with the same possibility of $1/4$ for moving up/down/left/right (i.e. random walk in 2D lattice). We define the stopping time 𝑇𝑐 as it hits $(a,b)$. How can we get the ...
3
votes
0
answers
516
views
The distribution of collision stopping time in 2D random walk
Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
1
vote
0
answers
176
views
Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
- 11
1
vote
1
answer
207
views
Expectation of the sum of the squares of the cardinal of an inverse function
I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as:
$$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$
where $\oplus$ is the bitwise XOR.
...
- 167
5
votes
3
answers
601
views
Convergence speed of a random dyadic rational generator
We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
- 1,677
6
votes
0
answers
156
views
Distribution of iid hypergeometric random variables conditioned on the sum
Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific,
$$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$
Let $S=X_1+\cdots+X_n$....
- 37.7k
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
3
votes
1
answer
153
views
Randomized version of Turán's theorem II
$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...
- 128k
5
votes
1
answer
209
views
Randomized version of Turán's theorem
Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...
- 128k
3
votes
1
answer
315
views
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
- 3,098
5
votes
0
answers
130
views
Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
- 83
1
vote
2
answers
163
views
Coupling a binomial - parity conditioning
If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (...
- 435
2
votes
0
answers
164
views
Finding an optimal strategy for a combinatorial sequential game
We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
- 83
1
vote
3
answers
339
views
Coupon collector targeting a collection among many
I am interested in the following problem:
We are given a universe $U$ of $n$ coupons, partitioned into $k$ collections, $C_1,\dots C_k$.
At each time step $t$, a coupon $X_t$ is selected uniformly at ...
- 111
2
votes
2
answers
379
views
Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$
Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that ...
- 1,677
0
votes
1
answer
340
views
Expectation of the ratio of two discrete random variables with combinatorial constraints
We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$.
...
- 1,677
3
votes
1
answer
395
views
Symmetric distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
- 1,677
2
votes
1
answer
218
views
Probability distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
- 1,677
3
votes
3
answers
2k
views
Probability of a given string being a substring of another string
I am in interested into the following problem. We are given an alphabet $\Sigma$ of $k$ letters and a fixed string $S_1$ of length $l$ defined over $\Sigma$. Given a probability distribution $D$ over $...
- 41
2
votes
1
answer
195
views
Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S
Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
2
votes
2
answers
185
views
Independence depth of linearly dependent random variables
Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise ...
- 2,618
2
votes
0
answers
85
views
How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?
Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...
- 267
4
votes
1
answer
272
views
How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?
Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...
- 267
1
vote
1
answer
134
views
Random optimization problem
Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...
- 1,677
1
vote
0
answers
221
views
Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?
Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...
- 53
2
votes
1
answer
106
views
How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set?
I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\...
- 1,372
8
votes
1
answer
171
views
On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
- 1,209
3
votes
1
answer
167
views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
- 33
4
votes
1
answer
648
views
What is the probability for a Binomial to be greater than other?
Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$.
What is the probability for $X$ ...
3
votes
1
answer
228
views
Density of a somewhat random set
The density of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.
Given a set of positive integers
$F= \{m_0<\cdots<m_{k-1}\}$,
let $C\subseteq \omega$...
- 909
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
3
votes
0
answers
116
views
Trace of Symmetric matrices in fixed rank
I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
- 179
2
votes
1
answer
266
views
A question about finite free convolution
For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
- 2,246
8
votes
1
answer
198
views
Tail bound of a distribution
Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$.
Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1})...
- 305
1
vote
1
answer
357
views
Does CLT hold for joint distribution of two dependent binomial variables?
Let $S_n$ and $T_m$ be two binomial variables satisfying $S_n\sim B(n,\frac12)$ and $T_m\sim B(m,\frac12)$. Define $\tilde{S}_n=\frac{2S_n-n}{\sqrt{n}}$ and define $\tilde{T}_m$ similarly. For any ...
- 111
13
votes
2
answers
669
views
An inequality for expected value of normally distributed variables
Question. Let $X_1,\dots,X_n$ be random variables with normal distribution. Is it true that
$$\mathbb E \prod_{i=1}^nX_i^{2k}\ge\prod_{i=1}^n\mathbb E X_i^{2k}$$for any $k\in\mathbb N$?
(The ...
- 4,535
6
votes
0
answers
133
views
Random Balanced Assignment
A balanced assignment from from $N$ objects to $K$ classes is a mapping $\sigma\colon \{ 1, \ldots, N\} \rightarrow \{ 1, \ldots, K\}$ such that
$$
\textrm{Card}( \sigma^{-1} \{j \} ) = \textrm{Card} ...
- 1,127
7
votes
3
answers
345
views
Concentration Bound of $0/1$ permanent
If I pick a random $0/1$ $n\times n$ matrix with $0$ occuring with probability $p$ then what does the distribution of the permanent look like?
- 13.9k
3
votes
1
answer
190
views
Solution for Moment problem
I want to invert a sequence of moments and find a function f(x) satisfying:
$$m_r=\int x^{r}f(x) dx=\int x^{r} dF(x)$$
The sequence of moments is given by:
$m_{2s+1}=0$
$m_{2s}=\sum_{k=1}^{s}\binom{...
- 333
10
votes
1
answer
263
views
q-versions of the geometric distribution and their names
I'm trying to set straight various $q$-deformations of the standard geometric distribution.
The geometric distribution on $\left\{ 0,1,\ldots \right\}$ is well-known, it has
$$
\mu_1(X=j)=(1-p)p^j,\...
- 1,765