On the mixture of some continuous distributions

In this note three aspects of the mixture of continuous distributions are considered. Firstly, a finite mixture of densities involving  [Formula Matemática] is considered and the distribution of the sum of independent variables, each from such a mixture, is obtained. Secondly, characterization of a mixture of k Gamma distributions is attempted. Filnally , non-central chi-square and doubly non-central F is discussed in relation to a finite mixture of normal distributions.