Question: Let (X) be a random variable with moment generating function (m(u)) and cumulant generating function (c(u)). Assuming that both functions exist, prove that [left.frac{d^{2}}{d u^{2}}

Let $X$ be a random variable with moment generating function $m(u)$ and cumulant generating function $c(u)$. Assuming that both functions exist, prove that

\[\left.\frac{d^{2}}{d u^{2}} \frac{m(u+\lambda)}{m(\lambda)}ight|_{t=0}=c^{\prime \prime}(\lambda)\]

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