Converting Normal distribution to Lognormal

Question

Let's say I have a random variable that is normally distributed with the mean of "a" and variance of "b". How can I get an equivalent lognormal distribution's parameters in terms of "a" and "b"?

Answer 1

The Log-Normal distribution is parameterized by the same parameters as the underlying Normal distribution. Specifically,

 

X ∼ N(a,b) ⇒ Y = e^X ∼ LN(a,b).

 

However, we can compute the mean and variance of the Log-Normal by noting the following:

 

E(Y^t) = E(e^tX) = M_X(t) = exp(at + bt²/2) is the MGF of the Normal distribution.

 

 

 

E(Y) = M_X(1) = e^a+b/2

 

 

Var(Y) = E(Y²) - [E(Y)]² = M_X(2) - [M_X(1)]² = e^2a+2b - e^2a+b = e^2a+b(e^b - 1).