# Existence of distribution for a Moment Generating Function

Question : Does a distribution exist for which $$M_X(t)=\frac{t}{(1-t)},|t|<1 ? \text{ If yes, find it. If no, Prove it.}$$

Answer : Since the mgf is defined as $$M_X(t)=\mathbb{E}e^{tX}$$, We necessarily have $$M_X(0)=\mathbb{E}e^{0}=1.$$ But $$\frac{t}{1-t}$$ is zero at $$t=0$$, therefore it cannot be an mgf.

Any moment generating function $$M$$ satisfies the property that $$M(0)=1$$. Since your proposed function does not satisfy this property, it can't be an MGF.