The probability-distributions tag has no wiki summary.
-1
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0answers
27 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
2answers
49 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
1answer
76 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 ...
25
votes
1answer
2k 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 ...
0
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0answers
42 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
...
0
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0answers
36 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
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0answers
32 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 ...
1
vote
1answer
55 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
1answer
50 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
1answer
53 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
1answer
96 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 ...
1
vote
3answers
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 ...
0
votes
1answer
60 views
Cramér-Wold device with limited angle and 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
1answer
47 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 ...
4
votes
2answers
152 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 ...
1
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0answers
33 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 $\mathbf{x}$ ...
7
votes
3answers
231 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
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0answers
66 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
1answer
46 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
...
3
votes
1answer
60 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
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0answers
45 views
Bounds or approximations for the conditional probability of an event involving correlated random variables
Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...
0
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0answers
28 views
Probability distribution of a function of a random variable $P(y(x))$ [migrated]
Do you have an idea about how to prove the following? (references will be useful)
Let $x$ a random variable with probability distribution $ρ(x)$, and $y=f(x)$ another random variable, then the ...
1
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0answers
24 views
Conditional Distribution of Inverse Wishart
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
1answer
80 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
3answers
193 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?
2
votes
1answer
123 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 ...
3
votes
2answers
171 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 ...
3
votes
2answers
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?
6
votes
1answer
380 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 $X$ ...
3
votes
1answer
89 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 ...
4
votes
0answers
51 views
Cramér-Wold theorem with independence assumption
Let $X = (X_1, X_2)$ be a random vector with joint probability density $p$. The celebrated Cramér-Wold theorem says that we can reconstruct $p$ from knowing the push-forward densities of $X$ under all ...
1
vote
1answer
54 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, ...
5
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0answers
204 views
1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
...
1
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0answers
36 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 ...
1
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0answers
49 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 ...
5
votes
1answer
171 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 ...
1
vote
1answer
71 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 ...
1
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0answers
49 views
What is entropy of a variable described by Knightian uncertainty? [closed]
I have asked this question at Theoretical Computer Science and received no response.
Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, ...
2
votes
3answers
152 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 ...
4
votes
3answers
356 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 ...
0
votes
0answers
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.
...
2
votes
0answers
52 views
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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0answers
100 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
$$A_n=\Big\{\frac{i}{n}:~ ...
1
vote
1answer
101 views
Euclidian norm of Gaussian vectors
Let $X \sim \mathcal{N}(0, \Sigma)$ be a Gaussian vector in dimension $N$. I am interested by the probability density of the random variable $\lVert X \lVert_2$.
If $\Sigma = {I}_N$, we recognize ...
1
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0answers
57 views
Cramér-Wold like theorem for independent random variables
Let $X$ be a random vector in $\mathbb R^n$ with probability distribution $\mathbb P_X$. Now when given only the family of distributions
\begin{align*}
\left\{ \mathbb P_{v_1 X_1 + \dots + v_n ...
4
votes
1answer
105 views
Does second order stochastical domination with increasing likelihood ratio imply first order domination?
This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing ...
4
votes
2answers
222 views
Estimate on gaussian distribution
Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...
1
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0answers
46 views
Characteristic function known on subsets
Let $X$ be a random variable in $\mathbb R^n$ with distribution $\mathbb P_X$. Given a (infinite) family of matrices $W_t \in \mathbb R^{n \times m}$ parameterized by $t \in \mathbb R$, suppose we ...
3
votes
1answer
74 views
Relaxing conditions for Cramer-Wold type theorem
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 ...
4
votes
1answer
191 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 ...
