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 $M_{X}(0) = 1$.
Moment Generating Functions
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probability
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1The inverse Laplace transform? – 2010-10-22
1 Answers
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$M_X(\log(t))$ is the probability generating function. Differentiate $k$ times and set $t=0$. Divide the result by $k!$.
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0$M_{X}(\log(t))$ is the probability generating function because $E[e^{x\log t}] = E[t^x]$? – 2010-10-24
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0Also the coefficient of $e^{kt}$ gives $P(X=k)$. – 2010-11-08