Paper 2, Section I, F

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

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

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

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

Let $Q$ be the probability that a single horse wins all $m$ races. Express $Q$ as a polynomial of degree $m$ in the variables $p_{1}, \ldots, p_{N}$.

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

*Typos? Please submit corrections to this page on GitHub.*