One interpretation of Jensen’s inequality is that randomization or dithering hurts, i.e., raises the average value of a convex function: Forf convex and v a zero mean random variable, we have E f(x0 + v) >= f(x0). This leads to the following conjecture. If f is convex, then the larger the variance of v, the largerE f(x0 + v).

(a) Give a counter example that shows that this conjecture is false. Find zero mean random variables v and w, with var(v) > var(w), a convex function f, and a pointx0, such that E f(x0 + v) < E f(x0 + w).