**4**

votes

**0**answers

101 views

### envelope function for a linear combination of gaussian distributions

Given a distribution $F$ defined as a linear combination of Gaussian distributions:
$F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$
I want to find a Gaussian function $Q = ...

**1**

vote

**1**answer

85 views

### Convergence to a k-dimensional Gaussian vector

Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law to a Gaussian random ...

**4**

votes

**3**answers

409 views

### Incremental entropy computation

After a quick internet search I found no method for incremental entropy computation.
Question 1
Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...

**2**

votes

**0**answers

93 views

### Marginalizing multivariate normal over defined interval

Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...

**0**

votes

**3**answers

287 views

### Random infinite sequence : Can machines generate truly random sequences. [closed]

Test : "A True Random Sequence Source and a computer producing a certain sequence of numbers are kept in separate rooms and judges try to tell them apart by conducting a series of tests on the ...

**0**

votes

**0**answers

41 views

### Statistics applied in A/B Testing and calculating ideal test duration for statistical significance

I'm trying to get my head around the maths used here:
http://visualwebsiteoptimizer.com/ab-split-test-duration/
Can someone point me in the right direction in terms of what algorithms/mathematics ...

**0**

votes

**0**answers

41 views

### Chi-square approximations for censored data

If you are working with censored data (that is, you only know a lower bound for the value of some observations), can you still use the Chi-square approximation for the statistic $D \equiv ...

**8**

votes

**0**answers

203 views

### Testing contrasts in statistics: Is this provably a hard problem, or not?

Scheffé's method for identifying statistically significant contrasts is widely known. A contrast among the means $\mu_i$, $i=1,\ldots,r$ of $r$ populations is a linear combination $\sum_{i=1}^r c_i ...

**1**

vote

**1**answer

68 views

### How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

Let $X=(X_{t},t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space S={1,...,K}.
Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random ...

**3**

votes

**0**answers

58 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 $g:\Bbb R\to \Bbb R$ ...

**0**

votes

**0**answers

87 views

### Orthogonal transformation on random matrices

Let $A$ and $B$ be two random matrices such that $A = U^T B V$. Here $U$ and $V$ are deterministic and orthogonal matrices. The random matrices $A$ and $B$ have the same distribution?

**1**

vote

**0**answers

52 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

224 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

98 views

### What is the Bahadur-Anderson Algorithm?

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

**0**

votes

**0**answers

61 views

### repeated application of binomial distribution on a set of random variables

I have four solutions which are termed: A1, A2, A3, A4. These are actually the results of a searching algorithm. I know that A1 is the best solution, A2 is next to A1, A3 is next to A2 and A4 is the ...

**1**

vote

**0**answers

162 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

**0**answers

30 views

### What is the reliable way to find the errors on the eigen values of a covariance matrix?

My question may be a bit lousy. Suppose we have a set of statistical variables X1, X2, .. Xn, and we have N independent samples. We can compute the covariance matrix of {X}. My question is: Is there a ...

**0**

votes

**1**answer

137 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

315 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

324 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

351 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 ...

**0**

votes

**0**answers

129 views

### Normal correlation theorem in Banach spaces

I am looking for a version of the Bayesian Linear Model in Banach spaces.
Background: the finite dimensional case
The following is a well-known theorem about normal correlation for ...

**3**

votes

**2**answers

268 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

391 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

106 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

118 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

101 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

173 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

320 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

223 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

49 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

206 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 ...

**2**

votes

**0**answers

86 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

141 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

398 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

236 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

149 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

377 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

426 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

399 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

112 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

169 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

86 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

**2**answers

425 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 ...

**1**

vote

**0**answers

173 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

125 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

146 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

310 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

288 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 ...