Mixture of Distributions

Abstract

Mixtures of measures or distributions occur frequently in the theory and applications of probability and statistics. In the simplest case it may, for example, be reasonable to assume that one is dealing with the mixture in given proportions of a finite number of normal populations with different means or variances. The mixture parameter may also be denumerably infinite, as in the theory of sums of a random number of random variables, or continuous, as in the compound Poisson distribution. The operation of Lebesgue-Stieltjes integration, $\int f(x) d\mu ,$ is linear with respect to both integrand $f(x)$ and measure $\mu$. The first type of linearity has as its continuous analog the theorem of Fubini on interchange of order of integration; the second type of linearity has a corresponding continuous analog which is of importance whenever one deals with mixtures of measures or distributions, and which forms the subject of the present paper. Other treatments of the same subject have been given ([1], [2]; see also [3], [4]) but it is hoped that the discussion given here will be useful to the mathematical statistician. A general measure theoretic form of the fundamental theorem is given in Section 2, and in Section 3 the theorem is formulated in terms of finite dimensional spaces and distribution functions. The operation of convolution as an example of mixture is treated briefly in Section 4, while Section 5 is devoted to random sampling from a mixed population. We shall refer to Theory of the Integral by S. Saks (second edition, Warszawa, 1937) as [S], and the Mathematical Methods of Statistics by H. Cramer (Princeton, 1946) as [C].