Find the mean and the variance

Question

Let X be N(μ,σ²) and consider the transformation X = ln( Y ) or, equivalently, Y = e^X. ( The random variable Y is said to have the lognormal distribution.)

a) Find the mean and the variance of Y by first determining E(e^X) and E[(e^X)^2], by using the mgf of X.

b) Let X be N( 10, 4 ) and Y = e^X. Find P( 6,000 < Y < 18,000 ).