cumulant_expansion

cumulant_expansion - Let X be a random variable for which...

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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 2 is the variance). We can expand the function ln(E[e tX ]), called the cumulant generating function of X, as a power series in t as follows. ln(E[e tX ]) = ln(E[e t + t(X– ) ]) = μt + ln(E[e t(X– ) ]) = μt + ln(E[ ]) because e z = n z n! = μt + ln(1 +
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