# Tagged Questions

**-1**

votes

**0**answers

32 views

### Maximum chi-square distance between norm vectors [on hold]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below.
$d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$

**1**

vote

**3**answers

106 views

### Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...

**3**

votes

**2**answers

84 views

### expectation of log(x+a) when X follows a beta distribution

Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?

**0**

votes

**0**answers

70 views

### Fitting distribution to spatial data

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below.
...

**4**

votes

**2**answers

222 views

### Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...

**5**

votes

**3**answers

214 views

### How do you call the problem of approximating a continuous distribution with a simple discrete distribution?

The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...

**0**

votes

**0**answers

62 views

### Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...

**2**

votes

**2**answers

273 views

### What is the maximum entropy distribution on the natural numbers?

On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution.
Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...

**0**

votes

**0**answers

44 views

### Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...

**1**

vote

**1**answer

188 views

### Order statistics of independent NOT identically distributed random variables

I want to find the p.d.f of the n-th order statistics from a set of independent, but NOT identically distributed random variables $X_1, \dots, X_n$ (the p.d.f. of the $X_i$'s is at hand)

**2**

votes

**1**answer

324 views

### Central limit theorem for $P(x)\sim 1/x^3$ distribution

I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...

**0**

votes

**1**answer

55 views

### Wishart random variables

I have a question about Wishart random variable. If X follows a Wishart distribution, then does X-Y follows a Wishart Distribution if Y is a Hermitian matrix?
Thanks.

**4**

votes

**1**answer

433 views

### What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?

$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) ...

**1**

vote

**0**answers

71 views

### Sum of non-identical categorical random variables

Is there a named distribution for the sum of non-identical categorical random variables?
When the categorical variables are i.i.d., the sum is a multinomial distribution. When the categorical ...

**1**

vote

**1**answer

99 views

### What is a likelihood kernel?

The paper, "The Multinomial-Poisson Transformation" by S. Baker (see http://www.math.ntnu.no/inla/r-inla.org/papers/multinomial-poisson.pdf) presents "likelihood kernels" for multinomial variables, ...

**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

**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 ...

**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 ...

**1**

vote

**0**answers

323 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 ...

**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 ...

**1**

vote

**1**answer

145 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 ...

**3**

votes

**1**answer

117 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 ...

**0**

votes

**0**answers

179 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 ...

**0**

votes

**2**answers

190 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

97 views

### power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind:
user1, timestamp1
user1, timestamp2
user1, timestamp3
user2, timestamp4
user1, timestamp5
...
These ...

**1**

vote

**1**answer

230 views

### Product of probability densities of the form x^{-t} exp (-ax)

I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, ...

**1**

vote

**2**answers

215 views

### Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...

**3**

votes

**3**answers

365 views

### What is the name for a non-normalized distribution?

For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...

**2**

votes

**2**answers

4k views

### Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...

**0**

votes

**2**answers

301 views

### Do these random variables follow Gaussian distribution?

Say that a random variable $X$ follows the Gaussian distribution $\mathcal{N}(\mu, \sigma)$. Then will the ceiling $\lceil X\rceil$, the floor $\lfloor X\rfloor$, and the rounding $\lfloor X\rceil$ ...

**0**

votes

**1**answer

147 views

### Copulas and marginals thereof

Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...

**2**

votes

**2**answers

262 views

### Scale random variables in a way they have equal probabilities of being minimal

I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...

**0**

votes

**1**answer

302 views

### Expectation of little o in probablity [closed]

If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) ...

**3**

votes

**1**answer

337 views

### Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...

**5**

votes

**1**answer

722 views

### Eigenvalue distributions of finite dimensional Wishart matrices

I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...

**5**

votes

**1**answer

197 views

### Is the maximum tree-path length distributed lognormally (in the limit) ?

Consider a full binary tree with $k>10$ levels. Let the lengths of individual edges in this tree be i.i.d. random variables with finite moments. Then total lengths of the $2^{k-1}$ source-to-sink ...

**0**

votes

**0**answers

95 views

### A name for this distributional condition?

I have come across a needed condition on a continuous probability distribution defined over $[0, \infty]$ and wonder whether it has a name. For a distribution with CDF $F$ and pdf $f$ I need that the ...

**3**

votes

**0**answers

143 views

### Iterated Kumaraswamy distributions

The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$.
Does anyone know any formulas or properties relating to iterations of this on itself, meaning
$$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$
If ...

**1**

vote

**3**answers

721 views

### Concrete examples concerning standard deviations and mean absolute deviations

Once again stackexchange is not responding to one of my questions (so far no comments, no answers, two up-votes). Hence this "crossposting":
Is there a simple (or not simple?) algorithm that will ...

**2**

votes

**1**answer

458 views

### What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?

For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...

**8**

votes

**2**answers

457 views

### Order statistics (e.g., minimum) of infinite collection of chi-square variates?

Hi everyone,
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...

**6**

votes

**1**answer

1k views

### Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?

General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...

**4**

votes

**1**answer

755 views

### Monotonicity of the hard EM algorithm.

Consider the problem where we want to find a maximum likelihood estimate of $\theta$, given $X$ and $$P_\theta(Y) = \sum_z P_\theta(Y,x)$$ where $x$ is a latent variable.
I know that the soft EM ...

**1**

vote

**1**answer

195 views

### Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance

I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...

**3**

votes

**1**answer

297 views

### skewness of a truncated distribution function

Let $Y$ be a real random variable and $a$ a real number. Let $X:=\min(a,Y)$. Is it true that the skewness of $X$ is an increasing function of $a$?

**1**

vote

**1**answer

783 views

### Proof of conditional copula relation to the marginal copulas

Hello
I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this article or equation (6.3) of this book. I've seen this in many documents discussing conditional sampling ...

**1**

vote

**1**answer

295 views

### “Bridging” uniform and “mass” distributions

Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my ...