- #1

- 22

- 0

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

Thank you!

- Thread starter samuelandjw
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- #1

- 22

- 0

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

- 2,967

- 5

[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

- 22

- 0

Thank you, Dickfore.

[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!

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

[tex]

\left|\frac{d x}{d y}\right|

[/tex]

expressed as a function of

- #5

- 22

- 0

Thx.

[tex]

\left|\frac{d x}{d y}\right|

[/tex]

expressed as a function ofy. For what values ofydo the above expressions make sense?

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