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Math Help - standard normal expectations

  1. #1
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    standard normal expectations

    Calculate E[e^Z], where Z is the standard normal random variable (with density function f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}).

    So am I to calculate \int _{-\infty}^{\infty}e^{f(x)} \cdot f(x) \,\, dx?

    If I'm approaching this correctly, that leaves me with a mess stickier than a barrel of molasses which I don't know how to handle at all. I mean, I know that the integral of f(x) is F(x) where F(x) is the cumulative distribution function for the standard normal distribution, but I don't know what that is explicitly nor how to calculate it, let alone what is being asked here. In lecture, we have used tables of values for \Phi (x), where \Phi (x) (I assume) is this F(x), to approximate definite integrals of f(x) but I do not see how that might help here...so lost.

    Suggestions, please?
    Last edited by mr fantastic; November 25th 2009 at 05:35 AM. Reason: Corrected a typo in the pdf (since it's meant to be standard normal).
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  2. #2
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    A different interpretation

    Your problem might be asking you to evaluate:

    <br /> <br />
\int _{-\infty}^{\infty}e^{x} \cdot f(x) \,\, dx<br />
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  3. #3
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by qmech View Post
    Your problem might be asking you to evaluate:

    <br /> <br />
\int _{-\infty}^{\infty}e^{x} \cdot f(x) \,\, dx<br />

    and instead of completing the square, blah blah, to make this a valid density

    NOTE that this is just the MGF evaluated at t=1.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    and instead of completing the square, blah blah, to make this a valid density

    NOTE that this is just the MGF evaluated at t=1.

    Thanks for your replies.

    I assure you both, however, that I am stating the problem correctly. In fact, there is another problem I have that requires computing a similar integral (which I can only take as evidence that E[e^Z] is not a typo). To further complicate things, MGFs have not been an addressed topic so I must further assume that a solution is intended to be MGF-free.

    I really do appreciate your replies, and if you have any further advice would love to hear it.
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