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

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.