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.
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Expected value of the log of the factorial of a poisson distribution
I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from - http://en.wikipedia.org/wiki/Factorial_moment. Is there a similar expression for ...
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What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
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Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain
Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance matrix, and oblique (=...
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Infinite product of finitely-additive probability measures
I'm looking for a reference for the existence of a finitely-additive product probability measure for an arbitrary family of finitely-additive probability measures. It's easy to prove, but I'd like to ...
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Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
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calculating how much to oversell given an acceptable risk (statistics)
I have a shared resource with a finite capacity (let's say 100), and I have usage data (2 hours average of samples taken each 20 seconds). I accept a risk of 10% per year to reach the capacity.
...
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Factorization of permutations.
Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Consider the following ...
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Maximal principle for stochastic heat equation
Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...
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a problem on DTMC
For a Markov chain $\lbrace X_n, n\ge0\rbrace$ with transition probabilities $P_{i,j}$, consider the conditional probability that $X_n = m$ given that the chain started at time $0$ in state $i$ and ...
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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 $G$
$x_1, ..., x_q$...
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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 ...
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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 ...
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Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
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Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
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Distance between probability amplitude functions
Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For $...
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Distribution of random vectors
Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1$).
A vector $u\in ...
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Why this two model have same probability distribution?
(1)
Consider the following method of generating a random tree with $n$ nodes.
First expand the root node into two branches.
Then expand one of the two terminal nodes at random.
At time $k$, ...
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90
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Potentials of class D
A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...
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Continuity of sample paths of stochastic processes
Dear all,
[Bauer, Probability Theory, Exercise 2 of Chapter 39] -->
http://books.google.de/books?id=w76IHsPHybcC&pg=PA339#v=onepage&q&f=false
gives the following characterisation for ...
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Calculating or estimating a combinatorial multivariate sum
Dear all,
I'm currently looking at a problem in which the following combinatorial product emerges:
$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 !}{(m_1-n_1)!}\frac{(m_1+m_2-n_1)!}{(m_1+m_2-...
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Conditioning over Conditional probability? (also: $\phi$-mixing sequences)
For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows:
$$
\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
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Percolation on infinite percolation clusters
Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond ...
user16782
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129
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Divisible Random Variables
Suppose I can write a positive, real valued random variable
$$ X = m_1 X_1 + m_2 X_2,$$
where $m_1$ and $m_2$ are i.i.d, $X_1$ and $X_2$ are i.i.d and moreover, the $X_i$ are distributed like $X$. ...
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Random Permutation with fixed cycle length.
Suppose $ S_{n,N} $ be the set of $n$ elements with $N$ many cycles where $N$ is proportional to $n$. $U_{n,N}$ is an element picked randomly from this. It is known that the length of any cycle cannot ...
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Entropy of Bernoulli walks on semi-groups.
Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...
- 1,640
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Generating Conditional Random Graphs
Let $G(n,p)$ be the usual random graph on $n$ vertices with each edge existing independently with probability $p$ (no self loops , or double edges not are allowed). I would like to simulate the ...
- 4,902
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A random walk with uniformly distributed steps II
The problem is a improved version of this problem,
A random walk with uniformly distributed steps
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "...
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Help with derivation of probability density of {event generation} & {event detection}
I would like to specify a new probability distribution that relates to an event of size M being produced by some process and subsequently detected.
Some assumptions :
1) If the event is detected ...
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Normalized correlation with a constant vector
I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: http://en....
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Incremental computation of a conditional entropy
Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
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Random Walk in $\mathbb{R}^n$
Have there been papers dealing with random walks in $\mathbb{R}^n$ that are not on the lattice $\mathbb{Z}^n$? Instead of walking in one of the directions possible in $\mathbb{Z}^n$ with probability $...
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Is a random walk sample path dense in a finite region with reflecting boundaries?
If I start a random walk in an $n$-dimensional box , say $[0,1]^n$, with reflective boundaries (i.e. the random walk is never permitted to leave the box), will its orbit eventually be dense in the box?...
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Bounds on CDF for the median of samples from an exchangeable distribution
Suppose $x_1,\dotsc, x_n$ are $n = 2k-1$ samples from an EXCHANGEABLE sequence, where the common marginal distribution is assumed UNIFORM on $[0,1]$. Let $x_{(1;n)} \le \dotsc \le x_{(n;n)}$ be the ...
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Distribution of uniform-normed random vector
What is the pdf of $\vec{Y} = \frac{\vec{X} }{\lVert \vec{X} \rVert_\infty}$ with $\vec{X}$ a random vector following a multivariate standard normal distribution (zero-mean $\vec{\mu} = 0$ and ...
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$A \perp B$ and $A+B\perp r\left( 2A+B\right)$ for some continuous function $r$. Is there such a triplet $\left( A,B,r\right) $ with non-constant function $r$?
Let $A$ and $B$ be independent continuous random variables with supports $ \left( -\infty ,\infty \right) $ and $r$ be a continuous function. In addition, $A+B$ and $r\left( 2A+B\right)$ are ...
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Use of a priori information
I'm reading a paper [R1] where the authors propose a MAP estimator for the phase noise and frequency offset. However equation (17), which I reproduce below, represents a challenging step for me and I ...
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Proving that an optimal solution "converges"
This question is a follow-up on a previous question I asked at:
Distances between and among points in a region
Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...
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Motivating the use of the theory rough paths in stochastic analysis
I am a final year undergraduate looking to do a PhD in stochastic analysis, perhaps with applications to problems in mathematical finance. On a potential supervisor's webpage, it says that one of his ...
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density for Gaussian gram matrices
Let $Z \sim \mathcal{N}(0,\Sigma \otimes I)$ (so the columns of $Z$ are distributed $\mathcal{N}(0, \Sigma)$) and $A = Z'Z.$ Is there a name for the distribution on $A$? Is the density known?
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Conditional prob of value given input is drawn from subset = conditioning over subset?
Hey guys, I have a pretty basic question that I want to be sure of. I'm taking a probability over an input selected uniformly at random from binary strings of length $l(n)$. I would like to compare ...
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Has this process been studied?
Take a Poisson process on $\mathbb{R}$ with intensity given by Lebesgue measure. Think of this as the measure $d\mu=\sum_{n} \delta(t-\xi_n )dt$ where $\xi_n$ are the points of the process. Now ...
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Bounding a stochastic process in terms of time to return to 0
I have a $\mathbb{Z}_+$-valued stochastic process $X$ in discrete time, which has unit jumps up or down. I know the following statement: there exists a random variable $\tau$, almost surely finite, s....
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Optimizing for a unique outcome of a probabilistic marriage problem
Let's say I have some number of individuals who are single, $(b_1, ..., b_N) \in B$, and for every possible pairing of two individuals, $b_i$ and $b_j$, I happen to know the exact probability that the ...
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A maximum likelihood -based ranking system
Given a collection of sample rankings, what is the best way to compile them into an aggregate ranking? (Don't worry, I'm working towards a well-defined question.) There are at least two obvious ...
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Marginals and Convex Sets
I am looking for a weak convergence of marginals result, in the following type of situation. References to 'related' situations are also very much appreciated.
I have a collection of affine ...
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Probability and information. The burrel-bucket-glass problem.
Suppose we have a barrel with three different kinds of marbles: red, green and blue. The probability to find a red marble in the barrel is R0, analogously the probability for green is G0 and for blue ...
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How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
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129
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A M/M/$\infty$ queue of depositors with compound interest
Hello, I'm trying to model a bank's liabilities using a queue. Suppose a bank begins with a cash reserve of $M$. Depositors are a M/M/$\infty$ queue; they arrive with rate $\lambda$ and deposit 1 ...
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Renewal function - duality
Let us consider a random walk $(S_n)_n$. One denotes the instants of records of $-S_n$ by $0=T_0 < T_1 < T_2 \cdots$. Then for all $k$ one sets: $H_k=-S_{T_k}$. Finally, one define $\tau$ as the ...
- 541
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Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
