In general, the class of mixtures of the family of normal distributions or of Gamma (Type III) distributions or binomial distributions is not identifiable (see ,  or Section 2 below for the meaning of this statement). In  it was shown that the class of all mixtures of a one-parameter additively-closed family of distributions is identifiable. Here, attention will be confined to finite mixtures and a theorem will be proved yielding the identifiability of all finite mixtures of Gamma (or of normal) distributions. Thus, estimation of the mixing distribution on the basis of observations from the mixture is feasible in these cases. Some separate results on identifiability of finite mixtures of binomial distributions also appear.
"Identifiability of Finite Mixtures." Ann. Math. Statist. 34 (4) 1265 - 1269, December, 1963. https://doi.org/10.1214/aoms/1177703862