Consider a Gaussian random variable X, i.e., X follows a normal distribution with mean and variance 2 : X N (, 2 ). There is a function of X: u(X) = e X, (6) where > 0 is a parameter.

a. Compare E[u(X)] and u(E[X]). Are they equal? If not, which one is bigger.

b. Compute E[u(aX)], where a is a choice variable.

c. Show that the optimization problem max a E[u(aX)] (7) is the same as the following one max a a 1 2 2 a 2 . (8) Hint: The probability density function for normal distribution with mean and variance 2 is f(x) = 1 2 e (x) 2 22 . The expected value of a function g(x), when x is a random variable with a probability density function f(x), is R g(x)f(x)dx.