I'm reading this question about the independence of spacings of order statistics of exponential distributions.
I don't quite get how to find the joint density of $(X_{(1)}, X_{(2)}, X_{(3)})$ and the density function of each $Y_i$. Do we need to compute the density of $X_{(i)}$ and to show $(X_{(1)}, X_{(2)}, X_{(3)})$ is independent in order to get the joint density of $(X_{(1)}, X_{(2)}, X_{(3)})$ ?
