So evaluating the first derivative of the moment-generating function at 0 gives us , the first moment about the origin, also known as the mean.
That gives us the second moment about the origin, which comes in handy when finding variance, since variance is equal to .
This will keep happening for the third, fourth, fifth and every derivative thereafter, provided the moment-generating function exists. And lest you think that’s just some technicality you never have to worry about, you should know the venerable t and F distributions do not have moment-generating functions. It turns out some of their moments are infinite. So there you go.
This entry was posted in Expectation on August 6, 2011 by Clay Ford.
Hey there,thank you for the brief explanation. I would like to ask you,why do we need the first term (Σe^(tx)(0) when we compute the second derivative of the MGF since everything with x is constant ?