All Questions
Tagged with pr.probability probability-distributions
1,384 questions
1
vote
2
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141
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Greater contribution in a sum of independent random variables
In section 2.4 (Summation of strictly stable random variables), page 54, of the book "chance and stability", Zolotarev, Uchaikin there is the following consideration :
The general relation ...
- 305
0
votes
1
answer
288
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Random sampling from modified Erlang distribution [closed]
I am tasked with randomly sampling from the following probability density function, which is a modified Erlang Function:
$$f(k,q,\nu)=\frac{(k q)^{k-1}}{[(k-1) !]^{v}} \quad \text { with } \quad q \...
- 111
5
votes
3
answers
601
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Convergence speed of a random dyadic rational generator
We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
- 1,677
2
votes
0
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50
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Log-concave probability measure with slowest decay
Let $X$ be a real valued random variable with log-concave distribution $\mu$. For each $x \in {\mathbb R}$, let $$
\phi_\mu(x)=\min\limits_{c\in{\mathbb R}}E[e^{c(X-x)}]
$$
be the minimal value of the ...
- 781
1
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1
answer
259
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Is the topology generated by the convergence of finite-dimensional distributions metrizable?
Let $\mathbf{D} := D([0,1]; \mathbb{R}^d)$ be the Skorokhod space (equipped with the Skorokhod metric) of càdlàg functions, and let $X = (X_t)_{t \geq 0}$ be its canonical process. The space of ...
- 309
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0
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156
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Distribution of iid hypergeometric random variables conditioned on the sum
Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific,
$$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$
Let $S=X_1+\cdots+X_n$....
- 37.7k
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1
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323
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What is the expected value of the sum of the k (out of a set of n) smallest normal random variables?
Given $n$ independent normally distributed random variables $X_1,X_2,...,X_n \sim N(\mu,\sigma)$. For any $k\leq n$, let $X_{(k)}$ be the k-th order statistics (i.e., the k-th smallest value). What is ...
- 105
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0
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769
views
sub-exponential type upper bound on the Poisson probability
I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
- 11
1
vote
1
answer
197
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Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables
Consider a set of i.i.d. (positive) random variables $\{X_i\}_{i=1}^N$. Each variable $X_i$ has a distribution with finite mean but infinite variance. In particular, if $P_{X_i}(x)$ is the P.D.F. of ...
- 305
3
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2
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508
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Distribution of a certain functional of iid $N(0,1)$ random variables
Suppose that $X_1,\ldots,X_n$ are iid $N(0,1)$ random variables. Consider the random variable given by
$$
\xi_n
=\Bigl|\frac1{\sqrt{n}}\sum_{t=1}^nX_t\Bigr|^2-\frac1n\sum_{t=1}^nX_t^2
=\frac1n\sum_{s\...
- 423
2
votes
0
answers
302
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
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96
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Limit of a linear discrete-time stochastic process with uniform noise
I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it.
Suppose we have a stochastic linear ...
- 1
2
votes
1
answer
1k
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measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
- 23
3
votes
1
answer
206
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Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
1
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1
answer
172
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Is it always possible to determine the distribution of a random variable given all its moments? [closed]
we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet
and would appreciate an explanation with examples.
- 11
1
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1
answer
114
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Upper bound on the ratio of Poisson CDFs [closed]
Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for $n=1,2,3,...$}$$ Observe that, the ratio is $>1$ & as $n \to \...
- 135
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votes
1
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401
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Joint distribution of dependent Gaussians and their product
Consider a pair of dependent zero mean unit variance Gaussians, $$X,Y \sim \mathcal{MVN}\left(\vec{0},\begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}\right).$$
Their product $Z:=X\cdot Y$ is ...
- 133
6
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1
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237
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Ordering preference for two zero mean Gaussian outcomes
Let $X\sim \mathcal{N}(0,1)$ be a standard Gaussian random variable. If we let $f_a(x)\triangleq\mathbb{E}[\max\{aX,x\}]$ for $a,x >0$, how to prove that $$f_a(f_b(1))<f_b(f_a(1))~~\text{for }0&...
- 171
3
votes
1
answer
153
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Randomized version of Turán's theorem II
$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...
- 128k
5
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1
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209
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Randomized version of Turán's theorem
Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...
- 128k
1
vote
1
answer
240
views
Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
- 5,405
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votes
1
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119
views
Can I express this random variable in terms of known distributions?
By computing the Laplace transform of the total length of a random tree (the nested Kingman coalescent tree with coalescence rates $\gamma$ for the individuals and $\gamma'$ for the species), we would ...
- 1
1
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0
answers
104
views
An Inequality of Expected Value of Random Variables
I encountered the following problem in my research:
Suppose there are $N$ random variables that are independent and identically distributed (IID). The probability density function (PDF) of these ...
- 11
0
votes
1
answer
126
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Perturbative approach starting from a probability distribution approximated form
I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic ...
- 305
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0
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72
views
Integration of fractional function over Rice distribution
Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as
\begin{equation}
\int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx
\end{equation}...
- 377
1
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1
answer
117
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Modulus of continuity of parameterizing Wasserstein
Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported ...
1
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0
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225
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Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
5
votes
1
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191
views
Probability of gaps between coordinates of a random point on the sphere
Let $X=(X_1,\ldots,X_n)$ be a point chosen uniformly at random from the sphere $S^{n-1}\subseteq \mathbb R^n$. Given $a>0$, what is the probability that $|X_1|^2-|X_i|^2\geq a$ for all $i>1$? Is ...
- 741
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0
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44
views
Small parameter expansion of probability density
I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align}
\dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\
\dot{y}&(t)=v_0\cos{\beta(t)},\tag{2}
\end{...
- 11
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0
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86
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What probability distribution is this?
Thank you in advance for any suggestions or feedback.
I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$.
I am interested in finding the Wasserstein (...
1
vote
1
answer
370
views
in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?
In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...
- 21
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0
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113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
2
votes
1
answer
476
views
Is total variation distance of normalized sum of random variables to Gaussian monotonic decreasing?
Let $X_1, X_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S_n = (X_1 + \cdots + X_n)/\sqrt{n}$ to be their normalized sum. Define $D_n$ ...
- 763
2
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1
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336
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Uniqueness of deconvolution after convolution?
I have the following question and I'd greatly appreciate any help!
Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$
...
- 31
3
votes
1
answer
315
views
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
- 3,098
2
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0
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174
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Random sets of points and hyperplanes in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$.
Consider the random ...
- 1,677
3
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0
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98
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Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
1
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1
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276
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Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$
I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post.
Question:
Let $X ...
- 135
2
votes
1
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350
views
Covariance/Correlation matrix of $n$ random variables with uniform marginal distributions
Let $X_1, \cdots, X_n \sim \mathrm{Unif}[0,1]$ be $n$ random variables, each with marginal distribution being a standard uniform distribution. I want to characterize the set of covariance matrices (or ...
- 205
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1
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266
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CDF of a log-concave discrete random variable
In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.
My questions:
What can we say about this in the discrete setting?. For ex: Is the ...
- 135
1
vote
1
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92
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Geometric sampling problem in the Euclidean space in high dimensions
Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...
- 1,677
23
votes
7
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5k
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What makes Gaussian distributions special?
I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions.
...
2
votes
0
answers
83
views
Concentration inequalities for sets
Assume that we have a random set $B$ which is constructed by selecting elements from $U = \{ X_1, \dots, X_n \}$ where $X_i$ are independent samples from Gaussians with means $\mu_i$ and variances $\...
- 143
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1
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188
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Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function
Let's denote
$F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and
$F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...
- 250
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votes
2
answers
80
views
Name of distribution of the parameter of a Poissonian
Consider a Poisson process $\hat{n}$ with with parameter $t$ and distribution
$$f_t(n) = e^{-t} \frac{t^n}{n!}$$
Now instead suppose to have a random variable $\hat{t} \in \mathbb{R}^+$ whose ...
- 503
2
votes
1
answer
268
views
Union bound probability of random union
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E_i\}_{i = 1}^N,$ with $E_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$
Is there ...
- 245
1
vote
1
answer
107
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Tail bounds on random series in Hilbert space
Tail bounds on random series in Hilbert space
Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$,
$n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
- 11
1
vote
0
answers
75
views
Why does Y. Moshe Vardi use this specific matrix when estimating source-destination traffic intensities with EM algorithm?
Sorry for the verbose title, but the question is super specific. If you happen to know a site better suited for these types of question, feel free to direct me.
The article to which I am referring to ...
- 151
10
votes
2
answers
488
views
A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
- 128k
1
vote
2
answers
197
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Sampling method for a specific distribution in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, where $d\ll n$. Given any point $\mathbf{p}$ on the unit $(d-1)$-sphere $\mathcal{S}$, we define
...
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