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

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.

For $0\le x\le 1$, integrate the joint probability density function over its allowed $y$-range:Outside this interval it is zero. Therefore the marginal density isIts integral is $12(1/2-2/3+1/4)=1$, also checking the original joint normalization. By symmetry $Y$ has the same marginal density.

From the marginal density,The mixed moment isThus the covariance isThe negative sign is consistent with the support constraint: a large value of one variable leaves a smaller available range for the other.

For $0<x<1$, the conditional density is obtained by dividing the joint density by the positive marginal density:Consequentlywhich integrates to one. Its conditional expectation isThis uses the usual continuous conditional-density version, defined by the density ratio at a point where $f_X$ is positive; conditioning does not require the event $X=1/2$ itself to have positive probability.