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

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Discussion Overview

The discussion revolves around finding the expected value E(1/(1 + e^Z)) where Z is a normally distributed random variable. Participants explore various methods to approach this problem, including analytical and numerical techniques.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving the problem and mentions known results for E(e^Z) and E(1/e^Z) related to lognormal distributions.
  • The same participant suggests using the moment generating function but indicates they became stuck in the process.
  • Another participant proposes using Gauss-Hermite quadrature as a potential method for approximation.
  • The initial poster questions the applicability of Gauss-Hermite quadrature for obtaining an analytical solution and expresses uncertainty about its effectiveness.
  • There is mention of using Taylor expansion as an approximation method, though the resulting expression is described as not being very nice.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the problem, with multiple competing views on potential methods remaining unresolved.

Contextual Notes

Participants acknowledge limitations in their approaches, including the challenge of finding an analytical solution and the complexity of the resulting expressions from approximations.

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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