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Math Help - Standard Deviation of transformed R.V.

  1. #1
    Junior Member
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    Standard Deviation of transformed R.V.

    Hello all,

    Suppose you have some random variable x that is distributed according to a pareto distribution f(x) = k mk x-k-1. And then you have a transformation y = c x where c is a constant. I want to find out the standard deviation of the natural log of y. I have been trying for several hours but have had no success. I have tried to do it the standard way E[x^2]-E[x]^2 but am quickly running into a mess that I can't deal with. Does anybody here have any idea how I can approach this problem? I really appreciate the help.

    Cheers,
    N
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  2. #2
    MHF Contributor
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    Re: Standard Deviation of transformed R.V.

    ive got two suggestions for you:
    Simplify
    since y = log c + log x; the sd of y does not depend on the value of c.

    so without loss of generality, consider the case c=1.

    Use MGFs
    with C=1, consider the MGF of y=logx
    M_y(t) = E \left(e^{ty} \right) = E \left(e^{t \ln x} \right) = E \left(e^{\ln x^t} \right) = E \left(x^t} \right)


    E(x^t) is just the underlying (non central) moment of X. since x follows a standard distribution, you can look this up.

    Since you now have a working expression for M_y(t) you can use standard methods to obtain the moments of Y from M_y(t). (assuming you can differenciate E(x^t) at t=0.)

    PS: i haven't actually done this, but i assume it is tractable.

    PPS: if this post makes no sense to you, then you probably have not learned "moment generating functions" yet, and youll have to solve your problem the old fashioned way.
    Last edited by SpringFan25; July 28th 2012 at 10:42 AM.
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