Past exam of the options I course of the University of Oxford / 2012 / M / M2 / b / i / Solution

The variables are not independent. Their joint support excludes $X>1/2$ and $Y>1/2$ simultaneously, since every point with positive joint density satisfies $X+Y\le 1$. Yet each event separately has positive probability: the density is strictly positive on open triangular subregions where $X>1/2$, and similarly where $Y>1/2$. ThusThis contradicts the event factorization required for independence of random variables. The product-shaped formula $24xy$ on its support does not undo the dependence created by the triangular support.