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Math Help - How do you derive Mean and Variance of Log-Normal Distribution?

  1. #1
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    How do you derive Mean and Variance of Log-Normal Distribution?

    Hey,

    I found out the probability function



    But I don't know how to derive E(X) and Var(X) without using integration.

    Thank you, 892king
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  2. #2
    Senior Member tukeywilliams's Avatar
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    Use the moment generating function of the normal distribution:  M_{X}(t) = e^{\mu t + \sigma^{2}t^{2}/2} to get the moments of the log normal. In particular, suppose that  X has a log normal distribution with parameters  \mu and  \sigma . Show that  E(X^n) = e^{n \mu + \frac{1}{2}n^{2}\sigma^{2}} . It then follows that  E(X) = e^{\mu + \frac{1}{2} \sigma^2} and  \text{Var}(X) = e^{2(\mu+\sigma^{2})}- e^{2 \mu + \sigma^{2}} .

    Also  X = e^{Y} where Y is normally distributed with parameters  \mu and  \sigma . You use the transformation theorem to get the pdf of the lognormal. Interestingly, the log normal distribution does not have a moment generating function.
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  3. #3
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    Quote Originally Posted by 892king View Post
    Hey,

    I found out the probability function



    But I don't know how to derive E(X) and Var(X) without using integration.

    Thank you, 892king
    Of related interest: The Lognormal Distribution
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by tukeywilliams View Post
    Use the moment generating function of the normal distribution:  M_{X}(t) = e^{\mu t + \sigma^{2}t^{2}/2} to get the moments of the log normal. In particular, suppose that  X has a log normal distribution with parameters  \mu and  \sigma . Show that  E(X^n) = e^{n \mu + \frac{1}{2}n^{2}\sigma^{2}} . It then follows that  E(X) = e^{\mu + \frac{1}{2} \sigma^2} and  \text{Var}(X) = e^{2(\mu+\sigma^{2})}- e^{2 \mu + \sigma^{2}} .

    Also  X = e^{Y} where Y is normally distributed with parameters  \mu and  \sigma . You use the transformation theorem to get the pdf of the lognormal. Interestingly, the log normal distribution does not have a moment generating function.
    Can this be construed as not using integration?

    RonL
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  5. #5
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    Use momentgenerating function

    Actually using the moment generating fuction is quite easy to derive the moments of the lognormal:

    E(exp(t*log(X)) = E(X^t)...(1)

    also because log(X) is normal:

    E(exp(t*log(X)) = exp(ut+st^2/2)...(2)

    Then from (1) and (2) substituting with t = n:

    E(X^n) = exp(un+sn^2/2)

    From here you can easily obtain the mean and the variance.
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