Existence of distribution for a Moment Generating Function

Question

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.

Could you please explain the answer kindly ?

Answer 1

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.

Answer 2

It is based on the definition of MGF. $M_x(t)=E(e^{tx}) \\$

$M_x(0)=E(e^{0})=E(1)=1$

and for this case, $M_x(0)=\frac{0}{1}=0$