Jensen's inequality states that for a convex function and a random variable with a finite mean, .
(a) Suppose that where is a positive integer, and is a random variable taking values with equal probabilities, and where the sum . Deduce from Jensen's inequality that
(b) horses take part in races. The results of different races are independent. The probability for horse to win any given race is , with .
Let be the probability that a single horse wins all races. Express as a polynomial of degree in the variables .
By using (1) or otherwise, prove that .