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Take $\lambda >0$ as the rate of the exponential distribution. By independence, the joint probability density of the original pair is ${\lambda }^2{e}^{-\lambda (x_1+x_2)}$ on $x_1,x_2>0$. Put $s=Y_1$, $r=Y_2$. The inverse change of variables isIts Jacobian determinant isUsing the absolute value in the change of variables formula, the transformed joint probability density isand zero otherwise. The support is the entire positive quadrant, and the factors are separately normalized:Integrating out either variable therefore yields these factors as the marginal densities. Factorization of the joint probability density proves independence, withThe sum has a shape-two gamma distribution, and the ratio has a beta-prime distribution with parameters one and one, independent of $\lambda$. Its cumulative distribution function is $r/(1+r)$ for $r>0$.