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The PDF of the exponential of a Gaussian random variable

  1. Jun 25, 2011 #1
    What is the PDF of the exponential of a Gaussian random variable?

    i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)?

    Thank you!
     
  2. jcsd
  3. Jun 25, 2011 #2
    The PDF for:
    [tex]
    Y = \exp{\left(\alpha \, X\right)}
    [/tex]
    where the PDF for [itex]X[/itex] is:
    [tex]
    \varphi_{X}(x) = \frac{1}{\sqrt{2 \, \pi} \, \sigma} \, e^{-\frac{(x - a)^{2}}{2 \, \sigma^{2}}}
    [/tex]
    is given by:
    [tex]
    \varphi_{Y}(y) = \int_{-\infty}^{\infty}{\delta(y - \exp(\alpha \, x)) \, \varphi_{X}(x) \, dx}
    [/tex]

    Use the properties of the Dirac delta function to evaluate this integral exactly!
     
  4. Jun 25, 2011 #3
    Thank you, Dickfore.

    I realize I can just use [tex]
    \varphi_{Y}(y)=\varphi_{X}(x) \frac{dx}{dy}
    [/tex]
    and the result is almost log-normal distribution pdf, but your method looks quite interesting.

    I do have a question, in your last integral, suppose [tex]y[/tex] is constant in the integral, then the result would be [tex]\varphi_{Y}(y)=\varphi_{X}(x=\frac{1}{\alpha}\ln{y})[/tex], which is not quite the same as the log-normal. I'm not sure if I have gotten it wrong.
     
  5. Jun 26, 2011 #4
    you had forgotten the Jacobian of the transofrmation:
    [tex]
    \left|\frac{d x}{d y}\right|
    [/tex]
    expressed as a function of y. For what values of y do the above expressions make sense?
     
  6. Jun 26, 2011 #5
    Thx.
     
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