Stats 2 question

Question

Let X and Y be i.i.d. r.v.’s with common pdf f(t) = ( λe^(−λt), t ≥ 0, 0, otherwise.

(a) . Find the joint pdf for U = X/(X + Y ) and V = X + Y .

(b) Find fU (u) and fV (v).

(c) Are U and V independent? Why?

Answer 1

U = X/X+Y

V = X+Y

Thus, UV = X/X+Y(X+Y) = X; V - UV = Y

dX/dU = V

dX/dV = U

dY/dU = -V

dY/dV = 1 - U

Thus, as f(x,y) = λ²e^-λ(x+y) 0 <= x,y <= ∞

f(u,v) = λ²e^-λ(V) |V U |

|-V 1 - U|

= λ²Ve^-λ(V) 0 < U < 1 0 < V < ∞ (Note that, for any value of V, U is able to range from 0 to 1, so the domain is "rectangular", also, necessary for independence)

U and V are independent.