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x is a normal distributed variable. then what is the expectation of ln(1+e^x).

i simulated this distribution and find that when x is N(0, 100), the mean of this function is around 4.1, and when x is N(-10, 100), the mean is around 0.9.

can anyone tell me the exact expectation of this function, when x is normally distributed?

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Of course it's easy to write down the answer as the integral of $\log(1+e^x)$ times the density of the distribution. This lets you compute to much greater precision than you can get by simulation, but offhand I don't see any reason to think the integral can be evaluated in closed form. I'd recommend asking about this at math.stackexchange.com instead. –  Henry Cohn May 3 '13 at 12:57
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closed as too localized by Henry Cohn, Andreas Blass, Carlo Beenakker, Andres Caicedo, Noah Stein May 3 '13 at 13:35

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1 Answer

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It is easy to obtain lower bounds on the expectation by noting that $$ \ln(1+e^x) \geq \max\{0,x\} $$ which simplifies the expectation extremely and leads to the result for $X\sim \mathcal{N}(\mu,\sigma)$ $$ E(\ln(1+\exp(X))) \geq E(\max\{0,X\}) = \frac{\sqrt{\sigma}}{\sqrt{2\pi}} e^{-\frac{\mu^2}{2\sigma}} + \mu \Phi_{\mu,\sigma}(0) , $$ where $\Phi_{\mu,\sigma}$ is the distribution function of $X$.

The bounds obtained by this formula are $\mathcal{N}(0,100) : 3.99$ and $\mathcal{N}(-10,100): 0.833$.

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