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The 123 theorem of Probability Theory and Copositive Matrices

Alexander Kovačec, Miguel M. R. Moreira, David P. Martins (2014)

Special Matrices

Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality...

The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces

Nguyen Van Huan, Nguyen Van Quang (2012)

Kybernetika

We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.

The gamma-uniform distribution and its applications

Hamzeh Torabi, Narges Montazeri Hedesh (2012)

Kybernetika

Up to present for modelling and analyzing of random phenomenons, some statistical distributions are proposed. This paper considers a new general class of distributions, generated from the logit of the gamma random variable. A special case of this family is the Gamma-Uniform distribution. We derive expressions for the four moments, variance, skewness, kurtosis, Shannon and Rényi entropy of this distribution. We also discuss the asymptotic distribution of the extreme order statistics, simulation issues,...

The importance of being the upper bound in the bivariate family.

Carles M. Cuadras (2006)

SORT

Any bivariate cdf is bounded by the Fréchet-Hoeffding lower and upper bounds. We illustrate the importance of the upper bound in several ways. Any bivariate distribution can be written in terms of this bound, which is implicit in logit analysis and the Lorenz curve, and can be used in goodness-of-fit assesment. Any random variable can be expanded in terms of some functions related to this bound. The Bayes approach in comparing two proportions can be presented as the problem of choosing a parametric...

The inverse distribution for a dichotomous random variable

Elisabetta Bona, Dario Sacchetti (1997)

Commentationes Mathematicae Universitatis Carolinae

In this paper we will deal with the determination of the inverse of a dichotomous probability distribution. In particular it will be shown that a dichotomous distribution admit inverse if and only if it corresponds to a random variable assuming values ( 0 , a ) , a ∈ ℝ + . Moreover we will provide two general results about the behaviour of the inverse distribution relative to the power and to a linear transformation of a measure.

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