Related papers: Comparing the $G$-Normal Distribution to its Class…
The $G$-expectation framework is a generalization of the classical probabilistic system motivated by Knightian uncertainty, where the $G$-normal plays a central role. However, from a statistical perspective, $G$-normal distributions look…
We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that…
The bivariate Gaussian distribution has been a key model for many developments in statistics. However, many real-world phenomena generate data that follow asymmetric distributions, and consequently bivariate normal model is inappropriate in…
This paper examines the classical matching distribution arising in the "problem of coincidences". We generalise the classical matching distribution with a preliminary round of allocation where items are correctly matched with some fixed…
In this paper, we show that the G-normality of X and Y can be characterized according to the form of f such that the distribution of {\lambda}+f({\lambda})Y does not depend on {\lambda}, where Y is an independent copy of X and {\lambda} is…
The properties of the normal distribution under linear transformation, as well the easy way to compute the covariance matrix of marginals and conditionals, offer a unique opportunity to get an insight about several aspects of uncertainties…
For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision…
Heisenberg's uncertainty principle is often cited as an example of a "purely quantum" relation with no analogue in the classical limit where $\hbar \to 0$. However, this formulation of the classical limit is problematic for many reasons,…
Judging the significance and reproducibility of quantitative research requires a good understanding of relevant uncertainties, but it is often unclear how well these have been evaluated and what they imply. Reported scientific uncertainties…
A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an…
In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with…
Modern generative machine learning models demonstrate surprising ability to create realistic outputs far beyond their training data, such as photorealistic artwork, accurate protein structures, or conversational text. These successes…
We investigate the structure common to causal theories that attempt to explain a (part of) the world. Causality implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalizing power of the…
The statistical distribution of the ratio of two normal random variables is characterized by its heavy-tailed nature and absence of finite moments. The shape of its density function is highly variable, capable of exhibiting unimodal or…
There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…
In probability theory, there is a tendency to treat one random variable with a given distribution as being just as good as any other. By and large this is fine because probability is (mostly) concerned with distributional properties of…
We analyze a supersymmetric system with four flat directions. We observe several interesting properties, such as the coexistence of the discrete and continuous spectrum in the same range of energies. We also solve numerically the classical…
Use is made of rigorous definitions for the terms normal, natural, and harmonic to reveal a number of unfamiliar aspects about them. The Gaussian distribution is not sufficient to determine who is normal, and fluctuations above or below a…
We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of covariates are either missing or completely inaccessible, and (ii) the variance of the measurement error is undetermined…
The classical and quantum dynamics for an n-dimensional generalization of the kicked planar (n=1) rotator in an additional effective centrifugal potential. Therefore, typical phenomena like the diffusion in classical phase space are similar…