Let $X_1, . . . , X_n$ be independent random variables from an exponential distribution with rate 1, $Y_i =\sum^{i}_{j=1} X_j$, and $Z_i = \frac{Y_i}{Y_{i+1}}$, $i = 1,\dotsc ,n − 1$, $Z_n = Y_n$. I want to show that the random variables $Z_1,\dots,Z_n$ are independent.
I don't know if this will help, but I've found that $Y_i$ follows a gamma distribution. Now I'm stuck.
Any help is appreciated.
