marginal density problem

Question

Let $m(t)$ be a moment generating function of some random variable. Which of the following are Moment generating functions of some other random variables?

a. $m(t)m(6t)$

b. $-3m(t)$

c. $\mathrm{e}^{-t}m(t)$

d. All of these

Answer 1

Consider $\xi$ such that $m(t) = Ee^{t \xi}$. Suppose that $\xi$ and $\eta$ are independent and have the same distribution. Then m.g.f. of $\xi + 6 \eta$ is equal to $$Ee^{t (\xi + 6 \eta)} = Ee^{t \xi}Ee^{6 t \eta} = m(t) m(6t).$$ Also m.g.f. of $\xi - 1$ is equal to $$Ee^{t (\xi -1)} = Ee^{t \xi} e^{-t}.$$ Any m.g.f. is equal to $1$ when it's parameter is equal to $0$. Thus a) and c) are m.g.f. and b) is not.