How to Find E(1/(1 + e^Z)) for a Normally Distributed Z?

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SUMMARY

The discussion focuses on calculating the expected value E(1/(1 + e^Z)) where Z is a normally distributed random variable. Participants mention that E(e^Z) and E(1/e^Z) follow lognormal and inverse lognormal distributions, respectively. Suggestions for solving the problem include using Taylor expansion and Gauss-Hermite quadrature, although the latter's applicability for an analytical solution is questioned. The conversation highlights the challenge of finding a neat analytical expression for the expected value.

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Hejdun
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Hi everyone,

I am stuck with this problem. I am looking for E(1/(1 + e^Z)) where Z is a normally distributed random variable.

I know that E(e^Z) and E(1/e^Z) follow lognormal and inverse lognormal distibution and the means of these distributions are standard results. Of course, is also easy to find E(e^Z + 1).

However regarding my problem, does anyone have a suggestion of how to proceed? I tried to use the moment generating function but got stuck...

Thanks in advance!
/Hejdun
 
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Sorry to bump this.

Still no ideas of how to solve this problem?

Of course, I can approximate it using Taylor expansion, but the
resulting expression isn't very nice.

/Hejdun
 
Maybe Gauss-Hermite quadrature will give you a decent approximation?
 
bpet said:
Maybe Gauss-Hermite quadrature will give you a decent approximation?

Yes, the integral may be evaluated numerically,
but I am looking for an analytical answer. I am not sure how the Gauss-Hermite quadrature would help for such a case.

Thanks.

/Hejdun
 

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