On the Modes of a Mixture of Two Normal Distributions

Abstract

This paper presents a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters μ1, μ2, σ1, σ2, p, is unimodal or not. The approach of the study is based on a simple transformation, which might be useful for further investigation of the properties of such mixtures. It is shown, by a geometric interpretation, that a mixture of two normal distributions is either unimodal or bimodal. For finding the modes, a simple iterative procedure is given, which always converges. A brief table of the modes is provided for some values of the parameters. Some conditions under which we have a unique mode are studied. It is shown that, for this purpose, a general sufficient condition is |μ1 – μ2| ≤ 2 min (σ1, σ2). For the special case in which μ1 = μ2 = μ, a necessary and sufficient condition is presented, and from that the simple sufficient condition |μ1 – μ2| ≤ 2σ is derived.