A linear transformation changes a random variable’s expected value in a predictable way – if y = ax +

b, then E (y) = aE(x) +

b. Hence for this transformation [say, h(x)] we have E [h(x)] = h[E (x)]. Suppose instead that x were transformed by a concave function, say, g (x) with g′ > 0 and g″ < 0. How would E [ g (x)] compare with g [E (x)]?

note: This is an illustration of Jensen’s inequality, a concept we will pursue in detail in Chapter 7. See also Problem 2.14