Some Remarks on Exponential Families

The following facts may serve to provide a feeling about how restrictive the assumption of an exponential family is. (a)A one-parameter exponential family in standard form with respect to Lebesgue measure is a location parameter family iff it is normal with fixed variance.(b) It is a scale parameter family iff it is gamma with fixed shape parameter.Both facts are known (see Borges and Pfanzagl 1965; Ferguson 1962; Lindley 1958) but may not have received as much attention as they deserve.Under the assumption of differentiable densities, short and elementary proofs are given.