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Suppose that $X$ is a random variable with all of its moments finite, given by $ \mathbb{E}(X^k) = k! , k \in \mathbb{N}$. If $X$ has a moment generating function, find the distribution of $X$.

I'm lost in here. If someone could give me a hint of how to solve it, I appreciate any help. Thanks in advance.

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Consider the moment generating function $$ E(e^{tX}) = \sum_{x=0}^\infty e^{tx} P(X=x) = \sum_{n=0}^\infty \frac{t^n}{n!} E(X^n) = \sum_{n=0}^\infty t^n = \frac{1}{1-t}$$ provided that $|t|<1$. This looks like the MGF for the Exponential distribution with parameter 1.

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  • $\begingroup$ Thank you very much, it is very clear. I did not use the fact of $f_X(x) = P(X=x)$. I was lost because I did not know the p.m.f. $\endgroup$ – Davshock Feb 23 at 0:55

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