Let X be a random variable for which the moment-generating function exists. Let denote its mean. For j = 2, 3, ... , let j = E[(X – μ)j] be the j-th central moment of X (hence 2is the variance). We can expand the function ln(E[etX]), called the cumulant generating functionof X, as a power series in t as follows. ln(E[etX]) = ln(E[et+ t(X–)]) = μt + ln(E[et(X–)]) = μt + ln(E) because ez =nzn!= μt + ln(1 +
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