Questions tagged [probability-distributions]
In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.
2,065 questions
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Gibbs distribution via rigorous counting? [closed]
Consider the function $Dist: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ (natural numbers include $0$), by defining $Dist(E,N)$ to be the size of the set
$$
\{ (s_1, s_2, \ldots, s_N ) \in \mathbb{...
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Consistency of a measure witnessing a strengthening of Freiling’s axiom of symmetry
I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\...
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Log concavity of a Gaussian function
Fix $t > 0$ and consider the map
$$
f(x) = \log \mathbb{P}\{|\sqrt{x} + Z| \leq t\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
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The density of the direction of a random interval in a convex body
Let $K\subset \mathbb{R}^n$ be a convex body. We uniformly choose two points $X,Y$ in $K$ and denote the direction of $X-Y$ as $u$, where $u\in \mathbb{S}^{n-1}$, and $f(u)$ is the density of $u$. We ...
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Compound distribution from stochastic timelike foliation
In a two-dimensional Minkowski spacetime patch with light-cone coordinates $(U,V)\in(0,1)^2$, consider the timelike foliation defined by
$$
V(U)=e^{s/\ln U},\qquad s>0
$$
Randomizing the global ...
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Localized Yamada–Watanabe
I wonder if a Localized Yamada–Watanabe theorem up to a stopping time exists. Here is more details:
Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space ...
- 696
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Partial order on a probability space denoting "more typical"
$\DeclareMathOperator\supp{supp}\newcommand\teq{\underset t=}\newcommand\tlt{\underset t<}$Let $(X, \mathcal{B}(X),\mu)$ be a probability space, where $\mathcal{B}(X)$ is the Borel $\sigma$-algebra ...
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Gaussian expectation on a triangle is one-to-one with any single vertex?
Consider a triangle $\Delta$ in the plane with vertices $A$, $B$, $C$, as well as a second triangle $\Delta'$ which differs from $\Delta$ only by a single vertex ($C'$ instead of $C$). Let $\gamma(\...
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Distribution of the range of a 2d random walk excursion
For SSRW on $\mathbb{Z}^2$, let $R$ be the number of distinct sites visited by a random walk starting at the origin before it revisits the origin. Can we say anything about the asymptotic behavior of $...
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Quadratic bound for the cumulant generating function of Dirichlet?
Let $X \in \mathbb{R}^n$ have the Dirichlet distribution $\mathrm{Dir}(1, \dots, 1)$. In other words, $X$ is uniform on the $(n-1)$-dimensional simplex.
Define the mean-zero random variable
$$
Z_\...
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Expressing a discrete Gaussian distribution over a lattice as a mixture of uniform distributions over subsets of the defining lattice?
Context: For any vector $\mathbf{c}, \mathbf{x} \in \mathbb{R}^n$, real $s >0$, let $$\rho_{s,\mathbf{c}}(\mathbf{\lambda}):=e^{-\pi\lVert (\mathbf{x}-\mathbf{c})/s \rVert^2}$$ be a Gaussian ...
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Under what assumptions is the capacity of a communcation channel not degraded by confining the source distribution to a manifold?
Given the random variables $\boldsymbol{\mathrm{x}} \in \mathbb{R}^D$ and $\boldsymbol{\mathrm{y}} \in \mathbb{R}^d$, where $d < D$, a source distribution $p(\boldsymbol{\mathrm{x}})$, and a ...
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Is the non-central Beta distribution $\mathrm{STP}_{\infty}$?
$\newcommand\TP{\mathrm{TP}}\newcommand\STP{\mathrm{STP}}\newcommand\SVR{\mathrm{SVR}}$This excerpt is from the book Testing Statistical Hypotheses by Lehmann and Romano.
A family of distributions ...
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Distribution of inner product of unit vector of ones with normalized Gaussian with non-zero mean
In this Math overflow answer by @Iosif Pinelis, he demonstrates the distribution of the inner product two independently uniformly distributed vectors on the sphere $S^{d-1}$. In particular, the ...
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Approximate square root of probability distribution
Consider a random variable $X \sim P_X$ and the following density:
\begin{align}
p_Y(y;\sigma) = c \left\| \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \frac{(y-X)^2}{2 \sigma^2}\right) \right ...
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