The probability-distributions tag has no usage guidance.

**5**

votes

**1**answer

445 views

### How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...

**5**

votes

**2**answers

345 views

### An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this:
$G(s) = e^{a(s-1)^2}=\sum s^np(n)$
I need first to do Maclaurin expansion of the exponential and ...

**5**

votes

**3**answers

495 views

### Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...

**3**

votes

**2**answers

346 views

### Measure concentration for law of large numbers

The classical law of large numbers states that
$$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$
for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm.
I was wondering whether is it possible to ...

**0**

votes

**1**answer

164 views

### Residual lifetime of heavy-tailed random variable

The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula
\begin{equation}R(t)=P[X_\text{res}\leq t] = ...

**0**

votes

**0**answers

69 views

### Dominating Poisson with parameter depending on a Bernoulli

Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value ...

**7**

votes

**1**answer

125 views

### Distribution of dropped objects

Consider small perfectly elastic spheres being dropped from a fixed height in R^3, bouncing and coming to rest on the horizontal R^2. Assuming a reasonable distribution of minor perturbations of the ...

**1**

vote

**1**answer

353 views

### Brownian motion of every point in the plane

Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does ...

**1**

vote

**1**answer

198 views

### Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows
$$X\to W\to Y,$$ and $$X\to Y\to W.$$
How to prove that there exist functions $f$ and $g$ such that
...

**4**

votes

**2**answers

124 views

### Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not ...

**1**

vote

**0**answers

34 views

### the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...

**1**

vote

**1**answer

116 views

### N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...

**0**

votes

**1**answer

198 views

### Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function.
Thanks!

**0**

votes

**1**answer

49 views

### finite mixture of order statistics

Let $F(u)$ be a n-degree polynomial continuous distribution function in $[0,1]$, with $F(0)=0$, $F(1)=1$, that is $F(u)=\sum_{i=1}^{i=n} a_i u^i$. My question is: is that kind of distributions ...

**2**

votes

**1**answer

287 views

### Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets

Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a finite subset of ...

**1**

vote

**0**answers

29 views

### Minimal rectangular confidence regions

For a given multivariate pdf $f$ (mainly the gaussian one) I'm looking to compute a minimal rectangular confidence region for a given level $\alpha$. For example, I would like to solve problems of the ...

**2**

votes

**1**answer

238 views

### Probability distribution of uAv…

Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix.
What is the distribution of $u^HAv$ ( or $||u^HAv||^2$)
where : u is a column vector of U. v ...

**2**

votes

**2**answers

263 views

### How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...

**2**

votes

**1**answer

135 views

### What are the generalized Gaussian probability laws that are infinitely divisible?

We consider the probability density, often called a generalized Gaussian density, $$p_{\alpha}(t) \propto \exp (- |t|^\alpha),$$
with parameter $0<\alpha<\infty$. For $p = 2$, we recognize a ...

**2**

votes

**1**answer

33 views

### probabilistic distribution of given data

let us consider following model
$$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$
we have three parameter ...

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vote

**1**answer

200 views

### Push-forward density as surface integral [closed]

Let $X$ be a random variable taking values in $\mathbb R^n$ with a probability distribution $\mathbb P$ that has a density $p$.
Consider further a linear mapping $\pi: \mathbb R^n \to \mathbb R^m$, ...

**7**

votes

**1**answer

157 views

### Distribution of entries of a doubly-sorted random matrix

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...

**2**

votes

**0**answers

43 views

### How to get the Expectation of the normalization of some log-normal-distributions?

Problem Definition:
Suppose that a random variable of multivariate Gaussian distribution $X \sim N(\Sigma,\mu)$, $\Sigma$ is the covariance matrix, and $\mu$ is the mean. For each $x_i$ from $X$, $x_i ...

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votes

**1**answer

52 views

### Using Marchenko - Pastur type Theorems on Regression Analysis

Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g_{p_n}(x) = \sum_{k=1}^{p_n} ...

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votes

**1**answer

87 views

### A question about the distributions of order statistics

Let $F_{k:n}(x)$ denote the distribution function of $k$th order statistic, i.e. $k$th lowest of $n$ i.i.d. draws from a smooth distribution $F$ with support $[0,\bar{x}]$. Then $F_{k+1}(x)-F_k(x) ...

**8**

votes

**1**answer

236 views

### Concentration of sum of powers of normals

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...

**-1**

votes

**1**answer

110 views

### How to compute the limit of skewness function?

The skewness function of a list of values is:
where
$m_k=\sum_{i=1}^N (x_i-u)^k$
$u=E[x]$
The image shows the meaning of this function related to the shape of the distribution of its x values ...

**3**

votes

**2**answers

303 views

### Approximating Probability Distribution by Sampling

Consider a discrete probability distribution over $n$ events. Assume that the probabilistic kernel is a black box, that is, we can only sample from it without knowing anything about the type or ...

**0**

votes

**1**answer

284 views

### Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...

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votes

**1**answer

3k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**3**

votes

**1**answer

208 views

### 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
...

**2**

votes

**1**answer

329 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 ...

**0**

votes

**0**answers

71 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 ...

**2**

votes

**1**answer

230 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
...

**3**

votes

**1**answer

79 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 ...

**0**

votes

**1**answer

76 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} ...

**0**

votes

**1**answer

159 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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vote

**2**answers

378 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 ...

**1**

vote

**1**answer

213 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 ...

**0**

votes

**1**answer

101 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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votes

**2**answers

298 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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votes

**0**answers

62 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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votes

**3**answers

720 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
...

**2**

votes

**0**answers

101 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 ...

**3**

votes

**1**answer

377 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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votes

**1**answer

186 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 ...

**0**

votes

**1**answer

204 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 ...

**4**

votes

**1**answer

226 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 ...

**0**

votes

**3**answers

229 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?

**3**

votes

**1**answer

223 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 ...