Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,027 questions
13
votes
1
answer
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What's the maximum entropy probability distribution given bounds [a,b] and mean?
What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?
For example:
if a=0, b=1, ...
- 233
4
votes
1
answer
540
views
Continuous Markov Process and Change of Measure
So I've been thinking about a problem for a little bit and I decided it's time to ask those that know more about the subject than I do. I've been working on some Stochastic Calculus (a new area of ...
- 63
0
votes
2
answers
521
views
Bounds for number of coin toss switches
I toss $n$ biased coins and I want to count the number of times you get a H followed by a T or a T followed by a H. I call these switches. So for example if I get HHTTHTHHHT then I have $5$ switches ...
- 11
1
vote
2
answers
4k
views
minimum of different independent Poisson random variables
Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.
Is there any closed form expression or at least a good approximation for ...
- 171
1
vote
1
answer
104
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When does the limit of moments of multivariate distributions determine the limit distribution?
Hello
I'm sorry if this question is trivial but I haven't been able to find an answer. I'm trying to show that a sequence of distributions on $\mathbb{R}^n$ converges to the normal distribution by ...
- 13
15
votes
2
answers
6k
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Distribution of inverse of a random matrix
I got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose I have a fat random matrix (i,e., $R$ has dimensions $k\times d$ where $k<d$) whose
...
- 151
1
vote
1
answer
291
views
Area under Gaussian sample path curve
Dear All,
We know that the coastline paradox is related with fractal dimension of a curve. Now I want to know how to estimate the area under a sample path curve of a Gaussian process:
Given a ...
- 105
11
votes
2
answers
2k
views
Balls and bins variation
How many balls have to be thrown uniformly at random into $m$ bins, such that with high probability $n_1, n_2, \dots, n_m$ are distinct numbers, where $n_i$ is the number of balls in bin $i$ ?
Is ...
- 711
1
vote
0
answers
126
views
Conditional probability, deviation from the uniform distribution
Let $N\in\mathbb{N}$ and $G$ the group $\mathbb{Z}/n\mathbb{Z}$.
Let $q< N$ and:
$a_1, ..., a_q$ pairwise distinct
elements of $G$
$b_1, ..., b_q$ pairwise distinct
elements of $G$
$x_1, ..., x_q$...
- 43
21
votes
1
answer
725
views
What is the best probabilistic estimate from below for a random polynomial on an arc?
I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k x^...
- 61.9k
3
votes
2
answers
543
views
Independence using reflecting brownian motion
Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, and thus $X$ and $Y$...
- 41
1
vote
2
answers
313
views
Apparently simple probability
Hello,
Let $x\in[0;1]$ and $(B_i)_i$ be events defined by $P(B_i)\leq x, \forall i$.
Furthermore, this inequality is independent of the other events $B_i$ but the events are not necessarily ...
- 43
6
votes
2
answers
559
views
Inequality involving the weak second moment
I want to ask the following probability inequality:
Is it true that for any random variable $X\ge 0$, we have
$$
\sup_{t>0}(t\mathbb E(X\mathbf 1_{X\ge t}))
\le
2\sup_{t>0}(t^2 \mathbb P(X ...
- 105
5
votes
1
answer
502
views
Measurability issues in the proof of Fujisaki, Kallianpur and Kunita for stochastic filtering
I'm currently looking over the proof(s) of the theorem of Fujisaki, Kallianpur and Kunita regarding the MRT-like characterisation of square integrable random variables measurable with respect to the ...
- 51
3
votes
0
answers
133
views
What distribution(s) of delays make(s) timing attacks hardest?
$H$ is (Shannon) entropy.
In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$...
user5810
2
votes
0
answers
141
views
Brownian motion above another one.
We define
$p_T(f):= \mathbb{P}(\forall_{s\leq T}B_s \geq f(s)-1),$
where $B$ is a Brownian motion such that $B_0 =0$ and $f:\mathbb{R}_+ \mapsto \mathbb{R}$ is some (continuous) function. I am ...
- 1,491
1
vote
0
answers
153
views
Sampling without replacement: probability for total successes from successes in sample?
Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...
2
votes
1
answer
324
views
Period of linear congruential generator
This is a cross-post of the unanswered question (the given answer turned out to be incorrect) https://math.stackexchange.com/questions/245591/period-of-linear-congruential-generator .
How can you ...
- 21
3
votes
0
answers
2k
views
derivative of conditional expectation
Suppose $H:\Omega\times X\mapsto Y$
for some borel subset $X\subset \mathbf{R}$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$.
Further suppose that $H$ is $C^1$ for each fixed ...
- 71
25
votes
3
answers
2k
views
Persistent homology of Gaussian fields in Euclidean space
If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
- 44.4k
2
votes
3
answers
532
views
Minimum 1st-neghbors distance between N random points on a ring
We have $N$ points randomly and uniformly distributed on a ring of length 1.
Let $d_i$ be the distance between point $i$ and its first neighbor.
We want to know the expected value of the smallest $...
- 23
2
votes
1
answer
396
views
Manhattan distance vs. absorption time on an unbounded integer lattice
Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...
- 173
-2
votes
1
answer
283
views
How to work with infinite random graph(s) ?
Hi,
In the case where we are dealing with an infinite random graph (RG with infinite nodes).
How do we model/work with notions like degrees, degree distribution ? How are they defined ?
Thanks!
- 167
0
votes
1
answer
752
views
transform a polynomial into another one upto a constant
I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
- 169
1
vote
1
answer
199
views
Backing into a distribution function from an infinite moment sequence
Let's say you are given that $E(X^n)$ = $\frac{n!}{((n+3!)/3!)}$ for a random variable $X$. So the first 4 moments are $\frac{1}{4}, \frac{1}{10}, \frac{1}{20}, \frac{1}{35}$, and so on. Is there ...
0
votes
1
answer
749
views
distinguishing random orthogonal matrix from Gaussian random matrix
Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ ...
- 145
14
votes
3
answers
9k
views
Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves
In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can ...
- 173
19
votes
2
answers
1k
views
Particles chasing one another around a circle
Two particles start out at random positions on a unit-circumference circle.
Each has a random speed (distance per unit time) moving counterclockwise uniformly distributed
within $[0,1]$. How long ...
- 151k
9
votes
2
answers
366
views
Iterating Random Matrix Operations
Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...
- 15.5k
8
votes
1
answer
2k
views
total variation distance between two solutions of SDE
Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
- 931
4
votes
1
answer
213
views
Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 101
5
votes
1
answer
140
views
Do distinct idempotent measures on finite binary systems have distinct supports?
Suppose that $(S,*)$ is a finite set equipped with a binary operation. Extend the binary operation to the vector space $V$ with basis $S$.
The set of probability measures on $S$, viewed as a compact ...
- 3,547
1
vote
1
answer
140
views
Equivalence between choosing a subspace and choosing its orthogonal
Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\...
- 13
8
votes
2
answers
1k
views
Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations)
Let $X_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X_i]$ and $S_n=\frac{1}{n}\sum_{i=1}^nX_i$. This question concerns the tail ...
- 325
2
votes
1
answer
835
views
An optimization problem, non complete bipartite graph and hungarian algorithm
I have two tables at my disposal, one work dataset and one reference dataset. Each dataset has got two columns, lets say these are fields A and B. I would like the rows in reference dataset with the ...
- 123
4
votes
2
answers
2k
views
How to solve a specific multivariate recurrence relation (or general ones)
How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
$$f:(\mathbb{N}\cup\{0\})\times(\mathbb{N}\...
- 143
4
votes
2
answers
399
views
Generic words of given weight
Suppose you have an alphabet with countably many letters. Every letter has a particular weight (for instance, as in the game of Scrabble). There are a total of $n^2$ letters that have weight $n$.
...
- 1,329
5
votes
1
answer
595
views
Additive energy of random sets
Given two random sets $A,B$ in a finite field (say $x\in A$ independently and with probability $1/2$), what is known about the additive energy $E(A,B)=|\{(a,a',b,b')\in A\times A\times B\times B: a+b=...
- 1,701
2
votes
2
answers
195
views
Extremal point and probability
Let $(X,\mathcal{F},\mathbf{P})$ be a probability space and $f \colon X \mapsto \mathbf{R}^n$ an integrable function. We assume that $f$ takes its values in a closed convex set $C$ of $\mathbf{R}^n$ ...
- 103
4
votes
1
answer
830
views
Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
- 141
4
votes
1
answer
327
views
Gaussian Valued Random Variables in Geometry of Banach Spaces
Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very ...
- 43
0
votes
0
answers
98
views
coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
- 295
1
vote
2
answers
520
views
inequality for coupling of measures
Let $X = \prod _{s \in S} \Omega_s$, with $\Omega_s$ finite and all the same, $S$ countable. Let $\mu_1$ and $\mu_2$ be two probability measures on the product space (not necessarily the product ...
- 295
11
votes
1
answer
1k
views
Integration over the orthogonal group
Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is:
$$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$
This is quite a tricky question:
(1) The first thought would go to ...
- 1,363
3
votes
1
answer
673
views
convex combination of two covariance estimates
I am interested in covaraince matrix estimation. In brief: I have two estimates of the covariance matrix, and now I want to form a bona fide convex combination of the two.
Background: I have studied ...
0
votes
1
answer
320
views
Simple markov chain problem
I know this is an easy problem, but I can't figure it out.
A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$.
...
- 39
3
votes
1
answer
216
views
Does martingale convergence hold for arbitrary time?
Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B_i\subset\...
- 1,701
1
vote
0
answers
199
views
Existence of multidimensional Levy process with dependent structure
Levy process is frequently cited recently. When we come to multidimensional Levy process, the components are usually assumed to be independent. Are there any examples on how to construct a Levy ...
- 11
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 111
1
vote
0
answers
116
views
Proving an asymptotic property regard the fraction of '1' and '0' in binary sequences [closed]
Hello,
Consider the set of sequences of zeroes and ones of length $N$ with $k$ ones (or, Np ones where $p=k/N$). We draw randomly and uniformly a sequence from this set.
I want to show that with ...
- 11