Geodesic
Some Characterizations of the One-Parameter Family of Probability Distributions
Canadian mathematical bulletin, Tome 9 (1966) no. 1, pp. 95-102

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We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family. 
Mathai, A. M. Some Characterizations of the One-Parameter Family of Probability Distributions. Canadian mathematical bulletin, Tome 9 (1966) no. 1, pp. 95-102. doi: 10.4153/CMB-1966-013-8
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     title = {Some {Characterizations} of the {One-Parameter} {Family} of {Probability} {Distributions}},
     journal = {Canadian mathematical bulletin},
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     year = {1966},
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