Find the joint distribution of $(\frac{X}{X+Y},X+Y)$

Question

This problem is about joint probability distributions.

Let $X,Y$ be i.i.d. random variables, each exponential $E(1)$. I wish to find the distribution of $(A,B)$ where $A = \frac{X}{X+Y}$, $B = X+Y$. Equivalently, $X=AB, \, Y=B(1-A)$.

From the Jacobian method,

$$J(A,B)=\begin{vmatrix} B & A \\ -B & 1-A \\ \end{vmatrix}=B$$

Answer 1

To check that $A\perp\!\!\!\perp B$ you can use another method...

Observe that

• $B=X+Y$ is Complete and Sufficient Statistic

• $A=\frac{X}{X+Y}$ is scale invariant statistic

Being the model also scale invariant, $A$ is ancillary.

Thus, applying Basu's Theorem, you have the proof

To apply Jacobian method is very easy because you immediately find

$$f_{AB}(a,b)=b e^{-b}=1\times b e^{-b}=f_A(a)\times f_B(b)$$

$0\le a\le 1$

$b\ge 0$