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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 ?

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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.

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