Suppose that \(X\) is a \(\operatorname{PoIsson}(\lambda)\) random variable, so that the moment generating function of \(X\) is \(m(t)=\exp \{\lambda[\exp (t)-1]\}\). Find the cumulant generating function of \(X\), and put it into the form given in Equation (2.24). Using the form of the cumulant generating function, find a general form for the cumulants of \(X\).

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E +0(t"), (2.24) i! c(t) = i=1 Kiti