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
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.
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.