2. Consider the following distribution for the uncertain quantity P, that indicates my probability that a flipped coin will come up heads. With probability 2/3, I believe that the coin is fair, (P = 1/2). With probability 1/3, I believe that P is drawn from the density 3p2, 0 < p < 1.

(a) Find the cdf F of P. Is F non-decreasing?

(b) Use the Riemann-Stieltjes integral to find 1 1 pdF(p) and p2dF(p). 0 0

(c) Use the results of

(b) to find Var (P).