Describe a random variable with MGF $\frac{1}{2-M(t)}$ where $M(t)$ is the MGF of $X$

Question

If the MGF of random variable $X$ is denoted by $M(t)$, find a random variable which has the MGF $$\frac{1}{2-M(t)}.$$

I have tried to solve this problem using the Taylor series, but it did not work well!

Answer 1

Answer:

$\frac{1}{2-M(t)}=\sum_{k=0}^\infty \frac{M(t)^k}{2^{k+1}}=\sum_{k=0}^\infty E[e^{t\sum_{n=0}^N X_n}|N=k]P[N=k]$.

where N is Geometric(0.5).