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Suppose one is given an arbitrary moment generating function $M_{X}(t)$. How would you determine $P(X=k)$ from this? We know that $M_{X}(t) = E[e^{tX}]$ and $M_{X}(0) = 1$.

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The inverse Laplace transform? – Qiaochu Yuan Oct 22 '10 at 19:31
up vote 2 down vote accepted

$M_X(\log(t))$ is the probability generating function. Differentiate $k$ times and set $t=0$. Divide the result by $k!$.

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$M_{X}(\log(t))$ is the probability generating function because $E[e^{x\log t}] = E[t^x]$? – PEV Oct 24 '10 at 4:18
Also the coefficient of $e^{kt}$ gives $P(X=k)$. – PEV Nov 8 '10 at 17:30

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