# 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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### Maximum of bounded expectations at a certain Borel set?

Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued bounded random variable with a distribution measured on the Borel space $(\mathbb{R}^n,\mathcal{B}^n)$. Let $f:\mathbb{R}^n\to\mathbb{R}$ denote ...
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### Conditional expectation values defined by expectation values

I asked this question a couple of days ago on Math.SE but without any echo (no upvotes, although I offered a bounty). But because I did it for oversight from a reputable/professional source I now ...
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### Harmonic mean or generating function, pick one?

Is there an obvious or a profound reason for the dearth of statistical distributions for which analytical harmonic mean and moment generating functions co-exist? Are there good examples (with finite ...
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### Berry-Esseen unit ball

How can I show that sampling a random vector from a uniform distribution over a $d$-dimensional unit ball is similar to sampling a random vector from a uniform distribution over $d$-dimensional ...
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### Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]

Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. As noted in previous questions, I want to define an average using a new measure and integral since I found certain aspects of the ...
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### Manifold structure of Gaussian mixtures

Fix $l$ a positive integer. Let $\mathcal{M}$ denote the set of Gaussian mixtures of the form $$\sum_{i=1}^l k_i \mu_i,$$ where $\mu_i$ is a non-degenerate Gaussian measure on $\mathbb{R}^k$ and ...
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### Metrics on the space of distributions in terms of p.d.fs

If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a nice" metric $d_{\rm smooth}$ ...
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### Can we write down the density of this distribution?

Simple version: I am looking for the density of the random vector $(X+Z,Y+Z)$, where $X,Y,Z$ are independent gamma random variables (with non-restricted parameters). Next step: Actualy, i'm looking ...
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### Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?

Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
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### Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
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### Prove or disprove the linearity of expectiles

For expectation (mean), there are many useful properties such as Linearity of Expectation: $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$ $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$ (The 2 equations ...
This is a neater version of a question I posted here, on which I'm also stuck. The problem: Say I have a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-...