1.

Prove Jensen's inequality: if v( x) is convex, then E[ v( X)] ? v(E[ X]). Use the following definition of convexity: a function v( x) is convex if, and only if, for every x ₀ a line l ₀( x) = a ₀ x + b ₀ exists, such that l ₀( x ₀) = v( x ₀) and moreover l ₀( x) ? v( x) for all x [usually, l ₀( ) is a tangent line of v( )]. Pay special attention to the case v( x) = x ².

2.

Also prove the reverse of Jensen's inequality: if E[ v( X)] ? v(E[ X]) for every random variable X, then v is convex.

3.

Prove: if E[ v( X)] = v(E[ X]) for every random variable X, then v is linear.

4.

A decision maker has utility function , x ? 0. He is given the choice between two random amounts X and Y, in exchange for his entire present capital w. The probability distributions of X and Y are given by

and

Show that he prefers X