A novel extension of randomly weighted averages

Abstract

We study a well-known problem concerning a random variable Z uniformly distributed between two independent random variables. A new extension has been introduced for this problem and fairly large classes of randomly weighted average distributions are identified by their generalized Stieltjes transforms. In this article we employ the Schwartz distribution theory for finding distributions of this extension; we also study some of their properties.

Characterizations of arcsin and related distributions based on a new generalized unimodality

“…Remark 3.4 Though this Theorem (3.3) only applies under some very specific assumptions, it is a wild generalization of Conjecture 1.1. Also, some results of [13] and [8,10] are special cases of Theorem 3.3.…”

“…Remark 3.6 In the moments method finding the desired distribution may need having sufficient information about the solution of the problem. Of course in using the method of [10] one should know the Stieltjes transform of the distribution in question, but in the moments method one can approach the solution by the guesses resulted from calculating the moments sequentially.…”

Randomly Weighted Averages: A Multivariate Case

“…For given random variables X 1 , • • • , X n the distribution of the stochastic linear combination Z = n i=1 W i X i is used for the problems in lifetime, stochastic matrices, neural networks and other applications in sociology and biology. Let X i (1 i n) be the lifetime measured in a lab and 0 W i 1 be the random effect of the environment on it; so W i X i X i and thus n i=1 W i X i is the average lifetime in the environment (see Homei (2015)). Recently, several authors have focused on computing the lifetime of systems in the real conditions.…”

“…Indeed, the randomly linear combination of random vectors have many applications including traditional portfolio selection models, relationship between attitudes and behavior, number of cancer cells in tumor biology, stream flow in hydrology (Nadarajah & Kotz (2005)), branching processes, infinite particle systems and probabilistic algorithms, and vehicle speed and lifetime (cf. Homei (2015), Rezapour & Alamatsaz (2014) and the references therein) so finding their distributions has attracted the attentions of numerous researchers.…”