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Let $\varphi(\theta)=\mathbb E[e^{\theta X_1}]$ be the moment generating function and $\psi(\theta)=\log\varphi(\theta)$ the cumulant generating function of $X_1$. We compute $$ \varphi(\theta) = e^{-\theta}\mathbb P(X_1=-1)+e^{\theta}\mathbb P(X_1=1)=\frac12\left(e^{-\theta}+e^{\theta}\right)$$ and $$\psi(\theta) = \log\left(\frac 12 ...



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