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

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The discussion centers on finding the expected value E(1/(1 + e^Z)) for a normally distributed variable Z. Participants note that while E(e^Z) and E(1/e^Z) are manageable due to their known distributions, the original problem remains challenging. Suggestions include using Taylor expansion for approximation, though the results are not aesthetically pleasing. Gauss-Hermite quadrature is mentioned as a potential method for approximation, but the original poster seeks an analytical solution rather than numerical evaluation. The conversation highlights the difficulty of deriving an exact analytical answer for this expectation.
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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