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.
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Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
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Which coupling of uniform random variables maximises the essential infimum of the sum?
Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$.
Question: Let $\mu_1, \dots, \mu_n$ ...
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Asymptotic unitary invariance of rank-one spiked Gaussian matrix
I'm working on some Random Matrix Theory related stuff for my thesis, and i've come across the following problem:
Consider a (normalized) spiked Wigner matrix $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{...
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Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
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Measurability of $X$ with respect to $Y$ in conditional probability distributions
Let $\pi$ be a probability measure on $\mathbb{R}^2$ with respective marginals $\mu$ and $\nu$ such that $(X,Y) \sim \pi$.
Notation:
$\pi_{X=x}$ be the conditional distribution of $Y$ given $X=x$,
$\...
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Echoes of the chord
Just a fun problem I thought of.
A man is playing a magical pipe organ - every chord is an integer number of decibals (dB) loud. The softest chord is $0$ dB. Every chord of $N > 0$ dB creates a ...
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Existence of Dirac measures in the context of joint and marginal distributions
Let $\pi$ be the joint law of $(X, Y)$ with marginal distributions $\mu$ and $\nu$. We assume that we have: for all $A \in \mathcal{B}(\mathbb{R})$ such that $\mu(A) > 0$
$$
\nu\left(\{y \in \...
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Analytical approaches to approximate probability density functions of multivariate random functions
Given a random multivariate function $f(x, y, z)$, where $x, y, z$ are independent and identically distributed random variables with a probability distribution $\rho(X)$, I aim to approximate the ...
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Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
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A question on Poisson approximation of number of secure rooks on a d-dimensional chessboard
This question was given in our first year undergraduate Probability I course.
In $d$ dimensions the lattice points $i = (i_1, i_2, \cdots, i_d)$ where $1\leq i_j\leq n$ may be identified with the “...
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Parameters of Wishart distribution and generalized inverse
I recently came across the Wishart Distribution and a few things are unclear to me.
The Wikipedia page for the Wishart Distribution says that if $G=[g_1 \vert \; g_2\vert \; \ldots \vert g_n]$ is a $...
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How do you prove the triangle inequality property for a metric on Gaussians?
$$d(P,Q)=\frac{\left(\mu_1-\mu_2\right)^2+2{\left(\sigma_1-\sigma_2\right)}^2}{\left(\mu_1-\mu_2\right)^2+2{\left(\sigma_1+\sigma_2\right)}^2}$$
How do you go about proving that the formula has the ...
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Does quadratic asymptotic growth imply log-Sobolev inequality?
Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$.
Does this imply that irrespective of any other ...
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Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
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Conditional expectation of two independently drawn uniformly distributed variables
I have two independent variables $a$ and $b$. The first one, $a$, has been drawn from a uniform distribution $[m, 1+m]$ while $b$ has been drawn from a uniform distribution $[0,1]$: here $m$ is any ...
