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Deriving the expected utility comes from an application of the moment generating function of a Normal random variable: Let $U(W)=-e^{-\lambda W}$ and $W=x \cdot \bar{p}$ where $\bar{p} \sim N(\mu, \sigma^2)$ be given as above and then further note that $W$ is Normally distributed with $W \sim N(\mu \cdot x, x^2 \sigma^2)$. See that: \begin{align} \mathbb{E}\...

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