reading a book on financial calculus (Baxter/Rennie) I stumbled across the following problem:
X is a normally distributed random variable
thus its pdf is
f(x) = 1/(2 * π * σ^2)^0.5 * exp [(-(x – μ)^2)/(2 * σ^2)]
('π' in the above formula is pi)
suppose that h(X) = S0*exp(X)
using the Law of the Unconscious Statistician (Law of the unconscious statistician - Wikipedia, the free encyclopedia)
E(h(X)) = ∫h(x)f(x)dx
we’re then supposed to arrive at
E(h(X)) = S0*exp(μ+0.5*σ^2)
(you may also refer to page 7 of the book on Amazon (look inside feature): Amazon.com: Financial Calculus : An Introduction to Derivative Pricing: Martin Baxter, Andrew Rennie: Books – just turn to page 7)
Whatever way I tried to integrate, I did not arrive at that result (yeah, you’ve guessed it, I ain’t very adept at maths). I even tried with Mathematica's online integration feature but it failed to produce that result.
Any (foolproof, step-by-step) clues?
Thanks a lot!