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We prove that the distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of independent copies of such random variables., Comment: 15 pages

This note introduces a new range of modified gamma and beta $k$ functions. The authors present new modified gamma and beta $k$-functions, first and second summation relations, various functionals, Mellin transforms, and integral representations. Furthermore, mean, variance and the moment generating function of a generalized beta distribution are obtained.

A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\geq3$. Previously, a closed-form formula had only been available for $d=3$., Comment: 5 pages

Recently, extensions of gamma and beta functions have been studied by many researchers due to their nice properties and variety of applications in different fields of science. The aim of this note is to investigate generalized inequalities associated with extended beta and gamma functions

The object of this paper is to investigate the certain results involving Bateman's matrix polynomials for integral index. We obtain some properties, integral representation and recurrence relations for hypergeometric matrix function. We introduce some matrix differential equations of the three order, integral transform and fractional integral formulas for hypergeometric matrix function by using the beta and Laplace transforms formula, Erdelyi Kober type fractional integral operators

Let $M_n=\max \left(X_1, X_2, \ldots, X_n \right)$ denote the partial maximum of an independent and identically distributed skew-normal random sequence. In this paper, the rate of uniform convergence of skew-normal extremes is derived. It is shown that with optimal normalizing constants the convergence rate of $\left(M_{n}-b_n\right)/a_n$ to its ultimate extreme value distribution is proportional to $1/\log n$., Comment: 19 pages

The Rosenbrock function is an ubiquitous benchmark problem for numerical optimisation, and variants have been proposed to test the performance of Markov Chain Monte Carlo algorithms. In this work we discuss the two-dimensional Rosenbrock density, its current $n$-dimensional extensions, and their advantages and limitations. We then propose a new extension to arbitrary dimensions called the Hybrid Rosenbrock distribution, which is composed of conditional normal kernels arranged in such a way that preserves the key features of the original kernel. Moreover, due to its structure, the Hybrid Rosenbrock distribution is analytically tractable and possesses several desirable properties, which make it an excellent test model for computational algorithms.