 Solution

The uniqueness theorem for moment-generating functions states: if two real random variables $V,W$ have finite moment-generating functions on a common open interval $(-\delta ,\delta )$ with $\delta >0$, andthen $\boxed{V\text{ and }W\text{ have the same probability distribution}.}$ Finiteness throughout a neighborhood of zero is a hypothesis; the identity $M_V(0)=M_W(0)=1$ alone carries no information.