Suppose that $X$ and $Y$ are independent random variables with Exponential$(1)$ distribution. How can it be shown that $$ \frac{X}{X+Y} $$ and $$X+Y$$ are independent?
2026-03-26 11:04:10.1774523050
Independence of functions of exponential random variables
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Hints for one possible approach:
Find the distribution of $X$ given that ${X+Y}=k\qquad$ (for positive $k$)
Find the distribution of $\frac{X}{X+Y}$ given that ${X+Y}=k$
Observe that this second conditional distribution does not change with different $k$