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Expected Value of a Function (Standard Normal)

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Let Z be a standard normal random variable, f is a measurable function and a is a constant.
    Show E[Zf(e^aZ)] = aE[e^aZ]E[f'(e^(a(Z+a))]


    2. Relevant equations



    3. The attempt at a solution
    I set up an integral from negative to positive infinity of

    z*f(e^aZ)*e^((-z^2)/2)*1/sqrt(2pie)*dz

    My 2 questions are can I combine the z from the pdf of the normal distrbution function into f(e^aZ) and does the z in the beginning of my equation require integration by parts to evaluation the integral.

    Thanks!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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