Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 218 / 3 / b / Solution

Let $w=N(U)/N(R)$, so $1-w=N(V)/N(R)$. Since class counts add,The function $g(p)=p(1-p)$ is concave function on $[0,1]$. Jensen inequality therefore giveswhich is precisely $Q\le 0$. Thus an axis-aligned split cannot increase the weighted empirical Gini impurity.