Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 3F (Probability)

For $m\ge 2$, $f^{\prime \prime }(x)=m(m-1)x^{m-2}\ge 0$ on $[0,\infty )$, so $f$ is a convex function; for $m=1$ it is linear and also convex. The equally weighted random variable has expected value $EX=N^{-1}{\sum }_i{x}_i=1/N$. Applying Jensen's inequality givesMultiplying by $N$ provesFor $m>1$ equality requires all $x_i=1/N$, by strict convexity; for $m=1$ equality holds for every admissible vector.

For a fixed horse $i$, independence of the race results gives probability $p_i^m$ of winning every race. The events that different horses win every race are disjoint, soThis is a homogeneous polynomial of degree $m$. Apply part (a) to the nonnegative numbers $p_i$, whose sum is one, to obtain $\boxed{Q\ge N^{1-m}}$. For $m>1$ the smallest probability occurs when every horse has winning probability $1/N$; for a single race $Q=1$ regardless of these probabilities.