The sum of two independent exponential-type random variables

Edward Martin Bolger

Vol. 18 (1966), No. 1, 31–35

DOI: 10.2140/pjm.1966.18.31

Abstract

Let X₁, X₂ be nondegenerate, independent, exponential-type random variables (r.v.) with probability density functions, (p.d.f.) f₁(x₁;𝜃), f₂(x₂;𝜃), (not necessarily with respect to the same measure), where f_i(x_i;𝜃) = exp{x_ip_i(𝜃) + q_i(𝜃)} for 𝜃 ∈ (a,b) and p_i(𝜃) is an analytic function of 𝜃 (for Re 𝜃 ∈ (a,b)) with p_i′(𝜃) never equal to zero on (a,b). If X₁, X₂ are neither both normal nor both Poisson type r.v.’s, then X₁ + X₂ is an exponential-type r.v. if and only if p₁′(𝜃) = p₂′(𝜃).

Mathematical Subject Classification

Primary: 62.10

Milestones

Received: 17 August 1964

Revised: 26 February 1965

Published: 1 July 1966