### A chain of inequalities for some types of multivariate distributions, with nine special cases

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A characteriyation of the Gamma distribution in terms of the $k$-th conditional moment presented in this paper extends the result of Shunji Osaki and Xin-xiang Li (1988).

Is the Lebesgue measure on [0,1]² a unique product measure on [0,1]² which is transformed again into a product measure on [0,1]² by the mapping ψ(x,y) = (x,(x+y)mod 1))? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables X and Y constancy of the conditional expectations of X+Y-I(X+Y > 1) and its square given X identifies uniform distribution either absolutely continuous or discrete....

Data that are proportions arise most frequently in biomedical research. In this paper, the exact distributions of R = X + Y and W = X/(X+Y) and the corresponding moment properties are derived when X and Y are proportions and arise from the most flexible bivariate beta distribution known to date. The associated estimation procedures are developed. Finally, two medical data sets are used to illustrate possible applications.

A characterization of geometric distribution is given, which is based on the ratio of the real and imaginary part of the characteristic function.

We describe a class of bivariate copulas having a fixed diagonal section. The obtained class contains both the Fréchet upper and lower bounds and it allows to describe non-trivial tail dependence coefficients along both the diagonals of the unit square.

In this note we give an elementary proof of a characterization for stability of multivariate distributions by considering a functional equation.