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Math Help - Covariance of Normal and Log-normal RV

  1. #1
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    Covariance of Normal and Log-normal RV

    I'm doing a problem and have reached a snag. I have two RVs X~N(m1,s1) and Y~N(m2,s2), with Cov(X,Y) sXY.
    I've reached a step where I need to know how to calculate Cov(X,exp{cY}) where c is a constant.

    Any help would be appreciated.
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  2. #2
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    Re: Covariance of Normal and Log-normal RV

    Hey Foyboy543.

    Recall that E[e^(tX)] is the moment generating function and the MGF has a known form for the normal distribution (Replace the t value with a c: they are after all just dummy variables).

    Also recall that Cov(X,Y) = E[XY] - E[X]E[Y] and we know that E[e^(tY)] = MGF_t(Y) so now we need to deal with E[Xe^(cY)]. If these are not independent or un-correlated, then you will need to resort to the definition of E[X*e^(cY)] using the integral formulation under a bi-variate distribution.

    Since you have all the information for the joint PDF, you can calculate the above to get a specific answer.
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  3. #3
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    Re: Covariance of Normal and Log-normal RV

    Thanks for your help Chiro.

    I actually started with E(X*e^Y) but figured that Cov(X,e^Y) would be easier to calculate through some relationship.
    The thing is that I'm really trying to find E(Xi*e^Y) for i=1,2,...,N where each Xi is normal and has some covariance with Y. I have some other relationships specific to the problem that seem to indicate I will need to know E(Xi*e^Y) in terms of Cov(Xi,Y). E(X) and E(e^Y) are of course easy, but are you sure there is no "simple" way to write Cov(X,e^Y) in terms of Cov(X,Y)? I'm sure that integration is not the expected approach. There's a possibility that I've approached the problem in the wrong way and don't need to calculate E(X*e^Y) to begin with, but I think I'm on the right track if I can just rewrite Cov(X,e^Y) in terms of Cov(X,Y).

    EDIT: I think I figured out how to do it. Apply Stein's Lemma! Then we can say Cov(X, e^Y) = Cov(X,Y)*E(e^Y) and everything is smooth sailing. I think.
    Last edited by Foyboy543; October 24th 2012 at 06:54 PM.
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  4. #4
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    Re: Covariance of Normal and Log-normal RV

    Can you explain what Steins Lemma says?
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  5. #5
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    Re: Covariance of Normal and Log-normal RV

    If X,Y~MVNorm then Cov(X,g(Y)) = Cov(X,Y)*E[g'(Y)] provided it exists. So we take g(y) = e^Y then g'(Y) = e^Y. And of course for my particular problem there are constants floating around, but that's not important.
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  6. #6
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    Re: Covariance of Normal and Log-normal RV

    Sounds good if you can use it: I'll have to remember that identity for later on.
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