Sum and proportion of two exponentials

Topic: Functions of random variables and limit theorems · more questions on this topic
‹ A normal approximation on an interval The mean of repeated voltage readings ›

Let X and Y be independent and exponential with rate 1. Set U=X+Y and V=X/(X+Y), so that X=UV and Y=U(1-V).

  1. Find the absolute value of the Jacobian determinant of (u,v)\mapsto(x,y).

  2. Find the joint density of (U,V).

  3. Show that U and V are independent, and name their marginal distributions.

  1. x=uv and y=u(1-v). The Jacobian matrix is \begin{pmatrix} v & u \\ 1-v & -u \end{pmatrix}, with determinant -u. Its absolute value is u for u>0.

  2. The joint density of (X,Y) is e^{-(x+y)}=e^{-u} on the positive quadrant. Multiplying by the absolute Jacobian gives f_{U,V}(u,v)=u e^{-u} for u>0 and 0<v<1.

  3. Integrating in u leaves f_V(v)=1 on [0,1]. Integrating in v leaves f_U(u)=u e^{-u} for u>0, which is \operatorname{Gamma}(2,1) in the shape–rate parameterisation. The joint density is the product, so U and V are independent. V is uniform on [0,1].