# 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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### Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...
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### Almost orthogonality of independent random vectors [closed]

If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$, then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$. How can I show from the above result ...
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### Minimizer for Mean-Variance Portfolio Optimization [closed]

Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set $$\{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ?$$
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### Given two probability density functions find a number that satisfies a given equation

I have a problem for which I either need a proof or a counterexample. We are given two discrete random variables $x_1$ and $x_2$ in $[0, n]$ where $F_1(x)$ is the probability of $x_1\leq x$, and ...
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### What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version. Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
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### Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
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### Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time?

In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
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### Second moment of ranks

Suppose vector $R$ is a random permutation of the integers 1 through $n$ such that $$\mathcal{P}\left(R_i = 1\right) = \pi_i,$$ for given vector of probabilities $\pi$. Moreover, assume a '...
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### Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
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### Large deviation of random walk

1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk $$S_i = \sum_{j=1}^iX_j$$ for $i=1,2,\ldots,n$. I am looking for "good" exponential upper bounds ...
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### Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...
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### Shannon problem

Since a few days, I try in my research to model / formalize a source of Shannon a little weird, and I can't do it at all. First of all, I explain to you its operating principle and then I describe it ...
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### Statistical models of functions

I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...
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Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&... 1answer 40 views ### Characterisation of a superset of the simplex Does there exist a nice description of the following set: A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace, \... 1answer 90 views ### Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator? I have a basic question about Gaussian measures on a Hilbert space: Let$\mu$be a non-degenerate Gaussian measure on a Hilbert space$(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ... 0answers 73 views ### How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process? In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ... 0answers 104 views ### Probability of degree$0$gcd between every pair of random homogeneous polynomials shifted by random primes? Take$n,d,B\in\mathbb Z_{>0}$with$d<n$and denote$\mathcal M_{n,d}$to be set of all total degree$d$monomials in$n$variables$x_1,\dots,x_n$with degree$\leq1$in each variable (... 1answer 102 views ### Is there a coupling that induces a given coupling via a transition kernel? Let$X,Y$be two measurable spaces,$\mu,\nu$two probability measures on$X$, and$\kappa$a transition kernel from$X$to$Y$. Define$\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$and$\tilde\nu(dy)=\...
Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?
Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to \$(x,y)\equiv t(a,...