 Two independent exponential variables conditioned on their sum

If $X,Y$ are independent exponential random variables with one common rate $\lambda >0$, put $U=X+Y$. The change of variables for probability densities $(x,y)=(u-y,y)$ gives $f_{U,Y}(u,y)={\lambda }^2{e}^{-\lambda u}$ on $0<y<u$. Its marginal density is ${\lambda }^{2}u{e}^{-\lambda u}$, so the conditional probability density of $Y$ given $U=u>0$ is $1/u$ on $(0,u)$. This is the shape-$(1,1)$ case of beta-gamma independence. In particular $Y/U$ is uniform on $(0,1)$ and independent of $U$. Equality of the rates is essential: different rates leave an exponential tilt across the conditional interval.