moment generating function can anyone help me to inform the name of any book where I can get the following theorem, or give some detailed hint to solve this one:
Let $X$ and $Y$ be two random variables, if the moment generating functions of $X$ and $Y$ are equal then the probability distributions of $X$ and $Y$ will be same. In other words, if $E(e^{tX})=E(e^{tY})$ for all $t$ real, then $X$ and $Y$ have the same distribution.
 A: One can look at the contrapositive: $$F_{X}(x) \neq F_{Y}(y) \implies M_{X}(t) \neq M_{Y}(t)$$
I think it would be easier to look at characteristic functions (i.e. $\varphi_{X}(t) = E[e^{it}X]$).
A: An even stronger result is Levy's continuity theorem. Your question is easier, look it up here. If your variable is continuous, then the characteristic function is the Fourier transform of the probability density function, so all you need to do is run the inverse Fourier transform. For the general case, see the reference.
A: This chapter (which I found immediately by googling for 'moment generating function proof') contains a proof in the discrete case and a sketch of a proof in the continuous case. It also says that the theorem holds only under the hypothesis that the random variables have finite range.
A: The idea is to reduce the problem to the well-known case of characteristic functions (cf. Yuval's or PEV's answer). For a complete proof (for moment-generating functions), see Theorem 4.9 here. Further, see Theorem 8.1 here, and property (M1) here.
