# Thread: Gamma variables and transformations

1. ## Gamma variables and transformations

If Y ~ N(m, s^2), find the distribution function and then the density function of U = e^Y. Either
from this or by using the moment generating function of Y, find the mean and variance of U.
(The random variable U is said to have a lognormal distribution.)

2. Originally Posted by sonitgarg
If Y ~ N(m, s^2), find the distribution function and then the density function of U = e^Y. Either
from this or by using the moment generating function of Y, find the mean and variance of U.
(The random variable U is said to have a lognormal distribution.)
There are several options. One approach is to find the cdf of U:

$\displaystyle F(u) = \Pr(U < u) = \Pr(e^Y < u) = \Pr(Y < \ln u) = \, ....$

Then recall that the pdf of U is given by $\displaystyle f(u) = \frac{dF}{du}$. To differentiate you'll need to use the chain rule and the Fundamental Theorem of calculus.

Of related interest: http://www.mathhelpforum.com/math-he...tribution.html