If X and Y are independent and identically distributed unifo

If X and Y are independent and identically distributed uniform random variables on (0, 1), compute the joint density of

Solution

(a) U=X+Y,V=X/Y, => X=(vu)/(v+1) & Y= u/(v+1), |J|=(-u)/(1+v)², then g(u,v)=u/(1+v)², 0<u<1+(1/v), 0<v<u-1. (b) U=X,V=X/Y, => X=U &Y= U/V & |J|=u/v², then   g(u,v)=u/v², 0<u<1, 0<v<u. (c) U=X+Y, V=X/(X+Y), =>X=UV & Y=U(1-V) & |J|=U, then g(u,v)=u, 0<u<(1/v), 1<v<1-(1/u).