On some new properties of the beta distribution
The paper considers new presentations of a random variable following the beta distribution as a finite or infinite product of independent random variables, each one following a specific gamma distribution. In addition a convergent infinite series of gamma variables is provided, as well as a modified version of the Mieshalkin-Rogozin, theorem. More specifically, the author derives the representation of the beta random variable as a product (finite or infinite) of gamma variables by using properties of the Mellin transform and the Gamma function. Thus a finite or infinite product representation of the gamma function allows for the use of the corresponding gamma random variables via the use of their Mellin transform. In the sequel, using such a representation, a specific series involving the logarithms of the Gamma variables is proved to be convergent. Finally a modification of a theorem given by Mieshalkin-Rogozin is used to find bounds for the distribution function of the beta distribution. The product representation, discussed previously, is useful for the derivation of this modified theorem.
scientific article; zbMATH DE number 1271605