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Math Help - Expectation of the logarithm of a normal random variable, E(ln(X))

  1. #1
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    Expectation of the logarithm of a normal random variable, E(ln(X))

    Hello everyone,

    the title essentially sums it up: let X\sim\mathcal{N}\left(\mu,\sigma\right). My questions are:
    1. Is there an analytical result for \mathrm{E}\left(\ln\left(X\right)\right)?
    2. If not, are there any appropriate approximations?

    This question looks so innocent, I expect it must have been treated somewhere. I could, however, not find it in the relevant literature so far.

    Note that
    E(\ln(X)) \propto \int \ln(x) \exp\left(- \frac{\left(x-\mu\right)^2}{\sigma^2}\right) \mathrm{d}x
    The variable transformation Y:=\ln\left(X\right) leads to (ignoring integration bounds)
    E(Y) \propto \int y \underbrace{\exp\left(- \frac{\left(e^y-\mu\right)^2}{\sigma^2}\right)}_{p_Y\left(y\right)  }\mathrm{d}y
    So, my questions could also be interpreted as for the expectation of the random variable Y that is distributed according to a pdf that is proportional to p_Y\left(y\right). Unfortunately, I could not find any results for such a distribution, too.

    Any help is greatly appreciated!

    Thardan
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  2. #2
    Grand Panjandrum
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    Re: Expectation of the logarithm of a normal random variable, E(ln(X))

    Quote Originally Posted by thardan View Post
    Hello everyone,

    the title essentially sums it up: let X\sim\mathcal{N}\left(\mu,\sigma\right). My questions are:
    1. Is there an analytical result for \mathrm{E}\left(\ln\left(X\right)\right)?
    2. If not, are there any appropriate approximations?

    This question looks so innocent, I expect it must have been treated somewhere. I could, however, not find it in the relevant literature so far.

    Note that
    E(\ln(X)) \propto \int \ln(x) \exp\left(- \frac{\left(x-\mu\right)^2}{\sigma^2}\right) \mathrm{d}x
    The variable transformation Y:=\ln\left(X\right) leads to (ignoring integration bounds)
    E(Y) \propto \int y \underbrace{\exp\left(- \frac{\left(e^y-\mu\right)^2}{\sigma^2}\right)}_{p_Y\left(y\right)  }\mathrm{d}y
    So, my questions could also be interpreted as for the expectation of the random variable Y that is distributed according to a pdf that is proportional to p_Y\left(y\right). Unfortunately, I could not find any results for such a distribution, too.

    Any help is greatly appreciated!

    Thardan
    You do realise that the expectation is not real don't you?

    (because of what you say after your transformation implies that you do not)

    CB
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  3. #3
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    Re: Expectation of the logarithm of a normal random variable, E(ln(X))

    Ouch, yes, that should be obvious - nevertheless, I didn't recognize it. I don't know at the moment if it that actually is a problem in my context (the expression appears within a norm), but I have to check... Thanks for the hint.
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