All Questions
Tagged with pr.probability probability-distributions
1,384 questions
8
votes
0
answers
422
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
- 399
1
vote
1
answer
288
views
Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given
As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...
- 23
3
votes
1
answer
251
views
Another large noise limit
Note: Here all processes take values in $[0, 1]$.
Let $W$ be a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Let $X$ be the solution to the SDE
$$dX_t = \sigma X_t \, dW_t$$...
- 6,215
3
votes
1
answer
142
views
How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance?
Let $a, b: \mathbb R_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int_s^t\frac{k(u,y)}{1+a(u)}...
user478492
2
votes
1
answer
256
views
Does taking minimum preserve density monotonicity?
Suppose $X$ and $Y$ are continuous random variables with a joint density function $f_{X,Y}$. Both $X$ and $Y$ are supported on $(0,1)$ and have continuous (can be assumed differentiable) and non-...
- 31
5
votes
1
answer
187
views
Existence of a joint distribution on Bernoulli variables with same probability of being pairwise different
Let $m\in\mathbb{N}$ and $p\in(0,1)$ be arbitrary. Is there a sequence $X_1,\dots,X_m$ of random variables with the following specs on their distribution:
Each $X_i$ is unbiased Bernoulli: $X_i\sim {\...
- 1,096
-1
votes
1
answer
297
views
The distribution of the sum of values from a normal and a truncated normal distribution
Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...
1
vote
1
answer
603
views
Hammersley-Clifford theorem
The paper spatial interaction and the statistical analysis of lattice systems by Besag (1974) presents an alternative proof for the Hammersley-Clifford theorem.
In order to prove the HM theorem, Besag ...
- 113
2
votes
0
answers
264
views
Prove or disprove a mutual information inequality
I have $n$ IID Bernoulli random variables denoted by $X_1,X_2,\ldots X_n$ with parameter $p$.
I am interested in knowing if the following inequality involving mutual information holds :
$\boxed{\max_{...
- 83
2
votes
1
answer
292
views
Mutual information between two discrete random variables
I have 2 IID random variables $X_1$ and $X_2$ with $Bern(p)$ distribution. I have another binary random variable $Y$ taking values in $\{0,1\}$.
I am interested in comparing the following 2 mutual ...
- 83
1
vote
1
answer
310
views
Compute inverse cdf of normal distribution [closed]
How can I compute inverse CDF of normal distribution using the central limit theorem on uniform distribution (u[0,1])
- 29
1
vote
0
answers
19
views
Сonditional characteristics with respect to a discrete random variable [closed]
160 asymmetrical coins participate in the first roll. In the second roll, only those coins on which the "eagle" fell out in the first roll participate. It is known that the probability of an ...
- 11
0
votes
1
answer
370
views
Closed-form CDF for bivariate normal distribution in point $(\Phi^{-1}(p),\,\Phi^{-1}(p))$
Let $\Phi(x)$ be a CDF of standard normal distribution and $\Phi^{-1}(p),\,p\in(0,1)$ its inverse.
It is evident that
$$
\mathbb{P}(X<\Phi^{-1}(p))=\Phi(\Phi^{-1}(p))=p,
$$
where $X\sim N(0,1)$.
Is ...
1
vote
0
answers
66
views
Strong Rayleigh measures conditioned on partial sum
Consider a binary random vector $X=(X_1,\ldots,X_n)$ with a strong Rayleigh distribution (i.e., its multi-affine generating polynomial is stable). It is well known that the law of $X$ remains strong ...
- 234
-1
votes
1
answer
159
views
Poisson distribution and conditional expected value [closed]
I have a task:
Lat's take independent variables $X_{i}$ with Poisson distribution $Poiss(a)$. Distribution of $a$ has density $p(a)=\frac{8}{3}a^3e^{-2a}, a\ge0$.
Calculate:
$E(a|X_1=3, X_2=2, X_3=5, ...
- 101
4
votes
0
answers
2k
views
Show that $\mathbb{P}[ a V\le Z| V+Z]=\mathbb{P}[aV \ge Z| V+Z] \text{ a.s.} $ iff $V=\frac{1}{\sqrt{a}}Z'$ where $Z'$ is standard normal
Consider a pair of independent random variables $(V,Z)$ where $Z$ is standard normal. Now suppose that the following equality holds: for a given $a>0$
\begin{align}
\mathbb{P}[ a V\le Z| V+Z]=\...
- 671
1
vote
1
answer
109
views
Distribution of weight of special type of random-matrix vector product?
Let $G$ be a matrix of dimension $k \times n$ sampled uniformly randomly from $F_2^{k \times n}$. It is a well known fact that $y = xG$ is uniformly distributed in $F_2^n - \{0\}$ for all $x \in F_2^k$...
- 111
0
votes
1
answer
376
views
Random variable is Big O in probability notation
Let $R_n$ be a random variable with values in $[0,1]$ and $nR_n$ converges to $\frac{1}{1+C} \chi_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-...
- 115
5
votes
1
answer
512
views
Concentration inequality for Hilbert space valued random variables
I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...
- 115
2
votes
1
answer
131
views
Fisher Information of variance of difference between random variable and gaussian
I'm reading through the following paper: https://arxiv.org/pdf/0704.1751.pdf
I'm stuck in the middle of page 8, at the statement:
$$E[||S(X)-S^*(X)||^2] = J(X) - J(X^*)$$
Where $S(X)$ is the score of ...
- 35
2
votes
1
answer
297
views
Examples of "almost" Ahlfors regular measures
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...
- 5,405
1
vote
1
answer
386
views
Expected value of a function of normal random variable
Suppose $X\sim \mathcal{N}(0,\sigma^2)$, find the expectation $\mathbb{E}\left[\frac{1}{(1+X^2)^a}\right]$ where $a$ is a fixed positive real number.
Is there an explicit formula for the above ...
- 161
0
votes
1
answer
243
views
Integral form of expectation with respect to complex random variables [closed]
Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E_h[g(h)] = \int_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of ...
- 13
3
votes
1
answer
179
views
On the convexity of certain set of random vectors
Let ${\cal X}$ be the set of pairs of random variables $(X,Y)$ with finite expectations. For constant $c\in[0,1]$, define set
$$
\{(X,Y)\in{\cal X}:\exists a\geq 0, \, b\geq 0 \text{ such that } E[\...
- 7,182
3
votes
1
answer
631
views
Does convolution of a probability distribution with itself converge to its mean?
Suppose we have a probability density function $f(x)$ with a finite support $[a,b]$. If we take the probability convolution of $\lambda f $ with $(1-\lambda)f,0 <\lambda<1$ recursively for many ...
- 143
5
votes
2
answers
193
views
Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
- 715
1
vote
1
answer
88
views
Stationary and limiting distributions
Consider a CT Markov Process $X=(X_t)_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting ...
- 203
6
votes
1
answer
1k
views
Variance of the norm of a random variable under finite-moment assumptions
There is the following exercise in Vershynin's book on High-Dimensional Probability.
Exercise 3.1.6:
Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ ...
- 163
1
vote
1
answer
332
views
Extension of measurable function from dense subset
Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$...
17
votes
1
answer
1k
views
Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
4
votes
0
answers
144
views
Exponential families closed under affine transformations
Let $(\Omega,\Sigma,\mu)$ be a probability space and let $\mathcal{M}$ be an exponential family of probability distributions for $\mu$ of the following form: There are $\varphi_1,\dots,\varphi_n:\...
- 5,405
1
vote
1
answer
93
views
Probability distribution of $\sum_i^n X_i - T$ where $\sum_i^nX_i <T<\sum_i^{n+1} X_i $
Let $X_{1}, X_{2}, \ldots, X_{n}$ be IID random variables with mean $\mu$ and variance $\sigma^2$. Let $S_n=\sum_i^{n}X_i.$
Let $T\gg1$ and define $\tau=T-S_n$ where $n$ satisfies the following ...
- 117
1
vote
1
answer
613
views
Integral of the product of a gaussian pdf and cdf
I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
- 11
2
votes
1
answer
150
views
Existence of a process on $\mathbb{R}^2$ that looks like two 'independent' brownian bridges $B_1(x)$ and $B_2(x)$ conditioned on $B_1(x)+B_2(x) > 0$
Consider any probability density function $f(x)$ that has mean zero variance one and say all finite moments. You may assume standard normal density if you like.
Given $a_1,a_2>0$, I consider two ...
- 123
2
votes
1
answer
308
views
Maximum nearest neighbor distance for a Poisson point process
Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, almost ...
- 3,098
1
vote
1
answer
2k
views
First hitting time for a drifted Brownian motion
While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question.
Take a continuous-time stochastic process $X_t$ and define the the ...
- 21
0
votes
1
answer
79
views
Visualization PDF of distribution defined by quantiles
How can I visualise PDF of distribution defined by quantiles, that I predict with my neural network? Now I'm passing quantiles to the histogram, but I don't think it is the correct way for visualising....
1
vote
0
answers
664
views
The distribution of hitting time in 2D-lattice random walk [closed]
Assume a particle at $(0,0)$ with the same possibility of $1/4$ for moving up/down/left/right (i.e. random walk in 2D lattice). We define the stopping time 𝑇𝑐 as it hits $(a,b)$. How can we get the ...
3
votes
0
answers
517
views
The distribution of collision stopping time in 2D random walk
Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
2
votes
2
answers
161
views
Determine the affine envelope of a random process's MGF
Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...
- 265
0
votes
0
answers
84
views
Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
2
votes
1
answer
136
views
Does higher volatility of SDE imply lower probability of staying positive?
Given two SDEs $X^1$, $X^2$ :
$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$
where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
- 1,334
1
vote
1
answer
226
views
Orthogonal transformation of multivariate Bernoulli-Gaussian distribution
Actually, I have asked this question in https://math.stackexchange.com/questions/4330127/orthogonal-transformation-of-multivariate-bernoulli-gaussian-distribution, but I think mathoverflow might be ...
- 21
1
vote
1
answer
193
views
Identity for special case of Markov chain
Consider $P(X,Y)$ discrete and $Z = f(Y)$ with $f$ deterministic. The function $f$ identifies a partition of the elements of the alphabet $\mathcal{Y}$ of $Y$. Each outcome $z \in \mathcal{Z}$ is a ...
- 189
1
vote
2
answers
111
views
Concentration bound for sum of indicators of maximum value of k combinations
Let $X_1, \dots, X_n$ be i.i.d. random variables distributed as $\mathrm{Exp}(\lambda)$ for some $\lambda > 0$ and let $t > 0$. For every combination $J$ of $k$ of these variables, we define $...
- 143
2
votes
1
answer
377
views
Extension of subcopulas to copulas
This question is about the extension of subcopulas to copulas, shown in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look backward and forward." ...
- 108
4
votes
0
answers
118
views
What is the least compressible probability distribution? (under entropy constraint, for an expected squared error metric)
This is a cross-post from cstheory after a week with no answers/comments; I'm hoping someone here may have some thoughts.
Consider a distribution $\mathcal D$ over the reals, a real parameter $H\in\...
- 618
10
votes
1
answer
700
views
Martingales converging in probability but not a.s
It is known that a random series
$$
\sum_{n\geq 1} X_n
$$
whose terms $X_n$ are independent converges a.s. if and only if it converges in probability.
Is it true that a martingale $(Y_n)$ converges a....
- 34.7k
2
votes
1
answer
138
views
Comparison between $\|X\|_2$ and $\|X\|_{2,1}$
For any real random variable $X$, define
$$\|X\|_{2,1}=\int_0^\infty \sqrt{\Pr(|X|>t)}dt.$$
This quantity (it is not a norm) appears in various problems, e.g. the multiplier central limit theorem (...
- 197
4
votes
1
answer
538
views
L_infinity norm of two gaussian vector
$X = (x_1,...x_n) \in \mathbb{R}^n, X \sim \mathcal{N}(O, \Sigma_X)$ and $Y = (x_1,...x_n) \in \mathbb{R}^n, Y \sim \mathcal{N}(O, \Sigma_Y)$ are two independent gaussian vectors.
If $\Sigma_Y - \...
- 65