**4**

votes

**0**answers

66 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

**1**

vote

**0**answers

54 views

### Distribution for probability of an incorrect inference based on a comparison of only two samples?

I'm trying to demonstrate the problems of how taking a sample and assuming it reflects the population accurately can be problematic.
Imagine say an urn with some large number of balls, black and ...

**1**

vote

**2**answers

249 views

### A machine learning application question

I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve.
I want to predict the nature of user activity on a ...

**0**

votes

**1**answer

122 views

### What is the Bahadur-Anderson Algorithm?

What is the Bahadur-Anderson Algorithm, and which book could one read to learn it?

**1**

vote

**0**answers

213 views

### Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...

**0**

votes

**1**answer

177 views

### How to combine correlated signals !? [closed]

Hi everybody
There are 11 signals:
S_main : The original signal
S1 ~ S10 : 10 signals that are correlated to S_main with different correlation coefficients (coeff1 ~ coeff10)
Now here's the ...

**2**

votes

**0**answers

393 views

### Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$.
Then we can ...

**1**

vote

**0**answers

368 views

### Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that
\begin{equation}
\Sigma_{i=0}^\infty f(i) = 1,
\end{equation}
where
\begin{equation}
f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...

**2**

votes

**1**answer

414 views

### correlation for three variables? [closed]

suppose we have three variables here, x,y, z
now, what we know is that the correlation between x and z is 0.6, the correlation between y and z is 0.65.
Here is the question, is there any formula to ...

**3**

votes

**2**answers

336 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

470 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

141 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

121 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

160 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

207 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

544 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

233 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

56 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

247 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

98 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 ...

**3**

votes

**1**answer

483 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

366 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

170 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

380 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: ...

**10**

votes

**3**answers

531 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

502 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

186 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

105 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

190 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

97 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

527 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

199 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

138 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

157 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

316 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

347 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

81 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

129 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

530 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

343 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 ...

**17**

votes

**3**answers

1k 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

138 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

300 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

366 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 ...

**2**

votes

**1**answer

309 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

192 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

77 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

136 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

152 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 ...