Basu's Theorem's application

Question

I can't solve a problem, for which I am suppose to use Basu's theorem. Suppose that $X$ and $Y$ are independent Exponential random variables with common parameter $\lambda$. I have to show that $X + Y$ and $\frac X{X + Y}$ are independent. Can you help me on this?

Thank you!

Answer 1

Note that $X+Y$ is a complete sufficient statistic of the mean $\lambda$ (twice the mean, actually) and $Z=X/(X+Y)$ is an ancillary statistic since its distribution does not depend on $\lambda$. From Basu's theorem, it follows that these statistics are independent.

More direct approaches are possible, for example to compute the joint distribution of $(X+Y,Z)$ using a simple change of variables formula.