The PDF of the exponential of a Gaussian random variable

  • #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!
 

Answers and Replies

  • #2
2,967
5
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!
 
  • #3
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!
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.
 
  • #4
2,967
5
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?
 
  • #5
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?
Thx.
 

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