Hey,

I found out the probability function

http://upload.wikimedia.org/math/b/3...ba6498ceec.png

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

Thank you, 892king

- Jun 24th 2008, 07:39 PM892kingHow do you derive Mean and Variance of Log-Normal Distribution?
Hey,

I found out the probability function

http://upload.wikimedia.org/math/b/3...ba6498ceec.png

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

Thank you, 892king - Jun 24th 2008, 08:21 PMtukeywilliams
Use the moment generating function of the normal distribution: to get the moments of the log normal. In particular, suppose that has a log normal distribution with parameters and . Show that . It then follows that and .

Also where is normally distributed with parameters and . You use the transformation theorem to get the pdf of the lognormal. Interestingly, the log normal distribution does not have a moment generating function. - Jun 24th 2008, 09:05 PMmr fantastic
Of related interest: The Lognormal Distribution

- Jun 25th 2008, 12:16 AMCaptainBlack
- Nov 10th 2009, 05:39 PMDiegoUse 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.