The relationship between moments of truncated and original distributions plus some other simple structural properties of weighted distributions

A simple relationship between expectations of positive functions of random variables from truncated and original distributions is pointed out. This is a special case of a relationship between moments of weighted and original distributions which is also discussed. Another topic is that of expressing weighted mixture distributions as mixtures of weighted distributions and vice versa. Finally, an error in some structural properties of weighted distributions given in the literature is identified and corrected. Research carried out while the author held a British Science and Engineering Research Council Postdoctoral Research Fellowship in the Department of Statistics, University of Birmingham, U.K.