Paper 2, Section I, F

Probability | Part IA, 2010

Jensen's inequality states that for a convex function ff and a random variable XX with a finite mean, Ef(X)f(EX)\mathbb{E} f(X) \geqslant f(\mathbb{E} X).

(a) Suppose that f(x)=xmf(x)=x^{m} where mm is a positive integer, and XX is a random variable taking values x1,,xN0x_{1}, \ldots, x_{N} \geqslant 0 with equal probabilities, and where the sum x1++xN=1x_{1}+\ldots+x_{N}=1. Deduce from Jensen's inequality that

i=1Nf(xi)Nf(1N)\sum_{i=1}^{N} f\left(x_{i}\right) \geqslant N f\left(\frac{1}{N}\right)

(b) NN horses take part in mm races. The results of different races are independent. The probability for horse ii to win any given race is pi0p_{i} \geqslant 0, with p1++pN=1p_{1}+\ldots+p_{N}=1.

Let QQ be the probability that a single horse wins all mm races. Express QQ as a polynomial of degree mm in the variables p1,,pNp_{1}, \ldots, p_{N}.

By using (1) or otherwise, prove that QN1mQ \geqslant N^{1-m}.

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