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Let $X_1, X_2$ be independent exponential random variables with common pdf $f(x)=\lambda\exp(-\lambda x), x>0$. How do I show that $Z=X_1/(X_1 + X_2) \sim U(0,1)$?

I know that $F_Z(z) = P(\frac{X_1}{X_1 + X_2} <z) = \int \int_A f(x_1,x_2)dx_1 dx_2$, and I know what $f(x_1,x_2)$ is, but I am unsure how to find the bounds on the integral. Any help with this would be greatly appreciated.

David
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    This is a special case of the (answered) problem at https://stats.stackexchange.com/questions/252692 with $n=2.$ By definition, integrals have no bounds (they are always integrals over the entire space and bounds are solely to indicate a factor of an indicator function in the integrand); so if you truly know what $f$ is, there will be no problem. – whuber Jan 21 '22 at 15:15

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