Bayes limits and modes of a mixture of two normal distributions

The existence of Bayes limits and of two modes in the mixture of two normally distributed components may be regarded as criteria to use these components for representing classes in unsupervised learning. A geometric interpretation given by G. Doetsch (1936) and J. Behboodian (1970) is suitable as well for graphical as for iterative numerical solution. The geometric interpretation is, to find the modes as intersecting points of a transcendent function and a parabola. After an improved transformation of the abscissa given the transcendent function becomes independent of the parameters of the mixture. After a proposal of E.R. Berger a fractional rational function with absolute values being always larger than those of the transcendent is found. From its sufficient conditions are derived that the mixture be bimodal. The Bayes limits may be easily interpreted in the graphic representation of the transcendent function and parabola as intersecting points of that parabola and a parallel to the abscissa. The reader may use tables of the modes in recent publication. Furthermore programs for iterative calculation of the modes are available.