**3**

votes

**2**answers

283 views

### Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...

**0**

votes

**3**answers

411 views

### Probability that one RV will exceed many others

Assume the $1 \times N$ vector
$\mathbf X = [X_1, X_2, \ldots , X_N]$
contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable ...

**1**

vote

**0**answers

115 views

### Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...

**1**

vote

**1**answer

120 views

### Transition time in finite voter model

I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
...

**1**

vote

**1**answer

120 views

### ROC curve with repeated measures

Hi, I have some repeated measures data, one measurement a day for three days in a row, and the measured variable looks normally distributed. I have two groups, the "really ill" and the "not ill after ...

**3**

votes

**1**answer

183 views

### Exact sampling from 2D Ising model where coupling is constant?

What progress has been made towards sampling from the 2D lattice Ising model with the following Hamiltonian:
$H=-J\sum_{\langle i,j \rangle}S_iS_j - \sum_i b_iS_i$
Where the first sum runs over all ...

**2**

votes

**1**answer

366 views

### Expectation of the trace of an inverse of a random matrix

Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. ...

**2**

votes

**1**answer

228 views

### The first eigenvalue of a branching process matrix

Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$.
We know that if the first eigenvalue if $M$ ...

**0**

votes

**1**answer

52 views

### On Variance Break detection

Say that the variance of a constant mean scalar stochastic process can take finite number of values. The problem is to detect the the point of break in variance as observation data comes in.
I tried ...

**2**

votes

**1**answer

215 views

### Estimator for sum of independent and identically distributed (iid) variables

This interesting question was asked at http://math.stackexchange.com/questions/231455/estimator-for-sum-of-independent-and-identically-distributed-iid-variables a while ago but got no answers. The ...

**3**

votes

**0**answers

92 views

### A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...

**0**

votes

**0**answers

165 views

### Spectral densities and their corresponding covariance functions.

Hey guys, I'm currently doing a course in stochastic processes and have come across something that has been wrecking my mind for a while.
So, let's say that I have some even, symmetric function ...

**3**

votes

**1**answer

417 views

### The average number of people that can sit on a bench of a given length.

Let me explain what I mean:
The width of the average person varies, perhaps with a normal distribution.
Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...

**0**

votes

**1**answer

303 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...

**0**

votes

**1**answer

158 views

### How many ways we know to join two line segments with a smooth transitional function?

This topic was created to discuss how many ways we know to create piecewise linear functions with smooth transitions between the phases. An alternative is presents by Bacon & Watts (1971):
the ...

**0**

votes

**1**answer

378 views

### Has the controversy about *fiducial distribution* been settled? [closed]

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature?
G.P. Klimov: ...

**9**

votes

**3**answers

437 views

### Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...

**15**

votes

**1**answer

429 views

### The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...

**0**

votes

**1**answer

129 views

### Expression for the square of the correlation of two Gaussian variables as an expectation value

Dear all,
I might just be blind, so forgive me if it is a trivial question. Given two normally distributed variables $x_1$ and $x_2$ (with zero mean), their correlation $c$ can be estimated from the ...

**2**

votes

**1**answer

104 views

### How to calculate average lifespan of a new population?

What's the best estimate one can make about the average lifespan of a new population?
For instance, let's say an alien kind of life came to earth and we're able to breed then. We then get 1000 aliens ...

**3**

votes

**1**answer

171 views

### Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...

**0**

votes

**0**answers

89 views

### Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...

**5**

votes

**3**answers

468 views

### Concentration of sum of pairwise squared Euclidean distances of random vectors

Let $X_1, \ldots, X_n $ be independent random vectors in $B(0, D) \subset R^d$ ($\ell_2$ ball of radius $D$ centered at the origin). I am trying to find the concentration of the following quantity ...

**2**

votes

**0**answers

180 views

### Is connected correlation/cumulant expansion additive?

Say X is a free field or a Gaussian random variable.
Then I want to analyse the connected correlation, $<(X + a (X^2 - \langle X^2 \rangle))^n>_c$
I think that for $n \geq 4$ there are no ...

**2**

votes

**1**answer

132 views

### Optimization problem

I'm trying to solve a very practical optimization problem and I think I hit a dead-end.
There are $N$ products ($N \sim 50$). Each product can have a price $p_i$ in range between 1 and 40 dollars. ...

**1**

vote

**1**answer

147 views

### Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.

**4**

votes

**2**answers

313 views

### Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...

**2**

votes

**1**answer

302 views

### Probability Density Optimization

I am working on an optimization problem which I am stuck on towards the end.
Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define ...

**1**

vote

**1**answer

78 views

### Estimate which random variable has highest expectation

Problem:
You are given a sample of size $m$ from $n$ independent normally distributed random variables. Expectations and standard deviations of the random variables are unknown. Estimate, which ...

**3**

votes

**1**answer

124 views

### Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...

**5**

votes

**2**answers

421 views

### Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...

**0**

votes

**0**answers

248 views

### Markov transition probabilities and negative binomial distribution.

A realization of a Markov process generates a sequence of interval lengths between transition from one state to another. A natural way of modeling the distribution of the lengths is as a negative ...

**8**

votes

**1**answer

611 views

### What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...

**1**

vote

**0**answers

133 views

### 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 $[-1,1]$. ...

**8**

votes

**0**answers

254 views

### Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...

**6**

votes

**2**answers

363 views

### Most inconsistent ranking

A matrix of $k$ rows and $n$ columns is filled with the numbers $1,2,\ldots,k$ such that the following conditions are satisfied:
Every column contain all the numbers form 1 to $k$ without ...

**0**

votes

**0**answers

182 views

### Expectation of sample variance

Hi all,
Just a quick question - I want to make sure I'm not missing anything obvious here!
I'm trying to evaluate $E(S^2 \mid \bar{X} = \bar{x})$, where $X_1,\ldots,X_n$ are i.i.d. Normal($\mu, \sigma ...

**2**

votes

**1**answer

266 views

### Derivative of a random process

Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t.
I consider applying a (stochastic)derivative operation to the random process. What is the ...

**2**

votes

**0**answers

186 views

### What machine learning algorithm is appropriate for predicting one time-series from another?

I have eye-tracking data on two subjects -- a teacher, and a student. It's in the form (x, y, time), so there is a series of these for each subject. What the teacher looks at influences what the ...

**2**

votes

**0**answers

42 views

### probability particle paths .two sets of parameters

Consider a paticle is going from a to b
At the first part of its journey is one set of mean and standard deviation for a normal distribution. Then it keeps going contingent upon a second normal ...

**1**

vote

**0**answers

73 views

### 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.
...

**5**

votes

**1**answer

131 views

### Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?

Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic?
If so, what are necessary and sufficient conditions ...

**1**

vote

**0**answers

131 views

### CIMDO Methodology

Hi guys, I'm doing a quantitative finance research project (which includes quite a fair bit of probability and statistics) that is based on the following papers:
"Systemic Risk and Sovereign Debt in ...

**1**

vote

**2**answers

266 views

### Uniform law of large numbers for martingale difference

Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in ...

**0**

votes

**2**answers

191 views

### Are all variables in a set of random variables independent if all pairs are independent?

If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...

**0**

votes

**0**answers

80 views

### Markov renewal process with failure?

I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail.
I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = ...

**2**

votes

**0**answers

196 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**5**

votes

**0**answers

152 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**0**

votes

**1**answer

74 views

### Taking the partial derivative of the t-CDF with respect to the degrees of freedom

I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in ...

**2**

votes

**1**answer

144 views

### Moments of random matrices - when are they finite

I need to evaluate the moment
$$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is
$$X=ZZ^{\ast},$$ where
$Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...