Open Access
September, 1948 Mixture of Distributions
Herbert Robbins
Ann. Math. Statist. 19(3): 360-369 (September, 1948). DOI: 10.1214/aoms/1177730200


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].


Download Citation

Herbert Robbins. "Mixture of Distributions." Ann. Math. Statist. 19 (3) 360 - 369, September, 1948.


Published: September, 1948
First available in Project Euclid: 28 April 2007

zbMATH: 0037.36301
MathSciNet: MR26110
Digital Object Identifier: 10.1214/aoms/1177730200

Rights: Copyright © 1948 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1948
Back to Top