Possible and Impossible Moment-generating function

Question

$$M_1(t)=0.5(e^t + e^{-t}) \quad t\in R$$

$$M_2(t)=0.5(e^t - e^{-t}) \quad t\in R$$

$$M_3(t)=2*e^t - e^{2t} \quad t\in R$$

It's easy to see that all of the moments of M1 are exists, because all of its derivatives are exists (there is a random variable s.t. M1 is its moment-generating function).

But, according to what I've seen, there is no random variable s.t. M2, M3 is it moment-generating function.

I've managed to derive M2 and M3, so I do not understand why they can not be moment-generating function, and would like an explanation on it.

Thanks!

Answer 1

M2(0) != 1.

Also - M2, M3 are none convex function, and there-for M2, M3 are not valid moment generating functions.