The probability-distributions tag has no wiki summary.

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### Quantiles moments and Convergence

QUESTION:
Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence
...

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**1**answer

320 views

### Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian?
In ...

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47 views

### Concentration bound for $f(w) = w \times \sin wz$

I need to find an exponential bound for $P(|S_n - \mu| > \lambda)$ where $S_n = \frac{1}{D} \sum_{i=1}^D w_i \sin w_iz$ for a constant $z$, $E(S_n) = \mu$ and $w_i$ are drawn from the normal ...

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**1**answer

219 views

### A calculation involving a uniform random variable quantile

THE PROBLEM:
Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to
...

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votes

**1**answer

73 views

### Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...

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**1**answer

74 views

### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...

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**1**answer

158 views

### Is any derivative of $f_1^x f_0^{1-x}$ w.r.t. $x$ integrable?

For $f_0$ and $f_1$ two continuos probability density functions on $\mathbb{R}$, by Hölder, I know that $f_1^x f_0^{1-x}$ is integrable on $\mathbb{R}$, where $0 \leq x \leq 1$. Let $l=f_1/f_0$, then ...

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295 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 ...

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**1**answer

176 views

### Question about characteristic function with independence assumption

Let $X$ be a random vector taking values in $\mathbb R^2$ with probability density $p(x) = p_1(x_1)p_2(x_2)$, i.e. the components of $X$ are independent.
Let $V$ be an open set in $\mathbb S^1$, the ...

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**1**answer

87 views

### one divided by (constant plus complex Gaussian) [closed]

Let $X$ be a circular symmetric complex Gaussian random variable with zero mean and unit variance.
Define $Y=\frac{1}{A+x}$ for some real-valued constant A.
What is the distribution of $Y$?
When is ...

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**2**answers

250 views

### Joint probability distribution as functions

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...

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**0**answers

58 views

### Angular distribution for Gaussian vector with non-zero mean

The angular central Gaussian distribution (ACG) is the distribution of $\frac{\mathbf{x}}{\|\mathbf{x}\|}$, when $\mathbf{x}\sim\mathcal{N}\left(\boldsymbol{0},\mathbf{A}\right)$, where $\mathbf{x}$ ...

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585 views

### Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...

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**0**answers

96 views

### Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics.
I am already ...

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**1**answer

269 views

### Random weighted selection without replacement

I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$.
Selection procedure
...

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**1**answer

181 views

### concentration of random matrices involving normal random variables

Define the random variable
\begin{align*}
A=|a_1|^2\mathbf{a}\mathbf{a}^*
\end{align*}
where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as ...

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**1**answer

155 views

### Conditional Distribution of Inverse Wishart [closed]

Suppose $\begin{bmatrix}
K_{11} K_{12}\\K_{12}^T K_{22}
\end{bmatrix}\sim\mathcal{IW}\left(\eta,\begin{bmatrix}
\Sigma_{11} \Sigma_{12}\\\Sigma_{12}^T \Sigma_{22}
\end{bmatrix}\right)$.
What is the ...

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**1**answer

156 views

### General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...

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**3**answers

221 views

### Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?

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votes

**1**answer

195 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...

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424 views

### Gradient descent-like optimization on a convex landscape with noisy sampling

This is a rewrite of the original positing (below), and is crossposted to ...

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**2**answers

167 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?

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578 views

### Mean of i.i.d Random Variables With No Expected Value

Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...

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**1**answer

135 views

### Variance of maximum of mixture of gaussians

Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some ...

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**1**answer

100 views

### Is there a simple closed form solution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution?

I have an exponential distribution with rate $\lambda$, where $\lambda$ is drawn from a Gamma distribution with shape and scale parameters $(k,\theta)$. I'd like to calculate an exact PDF for values, ...

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**0**answers

409 views

### 1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
...

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**0**answers

53 views

### Does this kind of integral equations have unique solution?

Suppose $f_1$ and $f_2$ are two probability density functions on support $[0,1]$ (i.e. $f_1(x)=f_2(x)=0$ for any $x\not\in[0,1]$). Let $\varphi(x)$ denote a known probability density function on ...

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63 views

### Distribution of the local time for reflected Brownian motion on the quadrant

If $W$ and $B$ are two independent Brownian motions, $X_t=W_t-\sup_{u\le t} B_u + 1$. I want to find the distribution of $S$ the first hitting time of $0$ by $X$. If someone could give me a direction ...

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1k views

### Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...

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**1**answer

278 views

### explicit expressions of the distribution of sums of i.i.d. logistic random variables

Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4...
The expressions given in "On the convolution of logistic random ...

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202 views

### Expected value of swaps

Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...

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votes

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620 views

### Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$

Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but ...

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### Can truncated/non-smooth distributions be used as priors/posteriors in Variational Bayesian methods?

Variational Bayesian methods can sometimes be a good alternative to Markov Chain Monte Carlo numerical evaluation of probability distributions. They do this, as I understand it, by approximating the ...

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115 views

### approximation of probability distribution

I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying
$$\int_{\mathbb{R}}|x|d\mu<+\infty$$
Set
$$A_n=\Big\{\frac{i}{n}:~ ...

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**1**answer

1k views

### Euclidian norm of Gaussian vectors

Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$.
If $\Sigma = {I}_N$, we recognize ...

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96 views

### Cramér-Wold like theorem for independent random variables

Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions
\begin{align*}
\left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...

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127 views

### Does second order stochastical domination with increasing likelihood ratio imply first order domination?

This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing ...

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votes

**2**answers

250 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 ...

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votes

**1**answer

246 views

### Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution

Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total ...

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**1**answer

91 views

### Question about infinite-dimensional BM

Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by
$$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$
where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...

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### “Smallest” event such that probability greater than a given value

Very briefly, consider the probability space $(\mathbb R^n, \mathcal{B}(\mathbb R^n),P)$. During a problem I am studying, I came to a point where i need to compute
\begin{equation*}
\begin{aligned}
...

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267 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 ...

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### Is this exponential family Gaussian?

Let $m$ be a positive measure on the real line. Assume that $\exp k(t)=\int\exp(xt)m(dx)<\infty$ for $a<t<b$ and that $\exp k(t)=\infty$ if $t\notin[a,b],$, with $-\infty\leq a<b\leq ...

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votes

**1**answer

251 views

### Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the ...

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**1**answer

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### Maximal component of a multivariate Gaussian distribution

Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the ...

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89 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 ...

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317 views

### Repeated draws from multinomial distribution

(This is a cross-post from Math StackExchange http://math.stackexchange.com/questions/609641/multinomial-distribution-sum-of-squared-probabilities)
Let $\vec X = (X_1, \dots, X_k)$ be a draw from a ...

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**2**answers

406 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 ...

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### existence of Markov operators not generated by transition probability function

Transition probability functions can always be used to generate Markov operators, correct? So is it correct to say that a Markov process is a collection of Markov operators? On the other hand, are ...

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### Estimating the mean of a bivariate distribution by randomly sampling variates and rounding them to integer coordinates

Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, ...