A random variable is a measurable function from an uncertainty space to the set of real numbers, which is used to model randomness. An uncertain variable is a measurable function from uncertainty space to the set of real numbers, which is used to describe uncertainty. However, randomness and uncertainty often simultaneously appear in a complex system. Uncertain random variable provides a useful...

The amplitude-constrained capacity of a scalar Gaussian channel is shown to be achieved by a unique discrete random variable taking on a finite number of values. Necessary and sufficient conditions for the distribution of this random variable are obtained. These conditions permit determination of the random variable and capacity as a function of the constraint value. The capacity of the same Ga...

The aim of this paper is to propose a convex risk measure in the framework of fuzzy random theory and verify its advantage over the conventional variance approach. For this purpose, this paper defines the quadratic deviation (QD) of fuzzy random variable as the mathematical expectation of QDs of fuzzy variables. As a result, the new risk criterion essentially describes the variation of a fuzzy ...

A ratio distribution or quotient distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known.taining to a variety of product and quotient forms of random variables. As the probability density function,p.d.f,of random variable X. pdf of a quotient of uniform random variables. Up vote 1 down vote favorite. Suppose are IDD rand...

Fuzzy random variable is a combination of fuzzy variable and random variable, and can characterize both fuzziness and randomness in the real world. The mean chance of a fuzzy random event is an important concept in fuzzy random optimization, just like the probability of a stochastic event in stochastic optimization and the credibility of a fuzzy event in fuzzy optimization. In fuzzy random prog...

This paper considers the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A mild restriction on the regular conditional distribution function of X given o(Y) is presented which ensures that E[Q,(X-g(Y))] is minimized for any cost function Q, that is nonnegative, even, and convex. Further, we show that given any real valued Bore1 measurab...

Notation. we conform to the convention that random variables shall be denoted by capital letters, whilst constants and observed values of random variables be denoted by lowercase letters. If X is a random variable with a continuous and strictly increasing (cumulative) distribution function over its range, then the unique upper γ point ofX will be denoted byXγ , i.e. Pr{X ≥ Xγ} = γ. The sole exc...

A methodology is introduced for the identification of a multi-modal real-valued random variable from a collection of samples. The random variable is seen as a finite mixture of uni-modal random variables. A functional representation of the random variable is used, which can be interpreted as a mixture of polynomial chaos expansions. After a suitable separation of samples into sets of uni-modal ...

Several variable shape parameter methods have been successfully used in Radial Basis Function approximation methods. In many cases variable shape parameter strategies produced more accurate results than if a constant shape parameter had been used. We introduce a new random variable shape parameter strategy and give numerical results showing that the new random strategy often outperforms both ex...

We will estimate the size of a random process (X t) t∈T by the size of another (simpler) random process (Y t) t∈T. Define a Gaussian centered process to be when X t is a mean-zero random variable. For a random variable X