Let $X_{1}, \ldots, X_{n}$ be independent exponential random variables with parameter $\lambda$, and let $X_{(1)}, \ldots, X_{(n)}$ be their order statistics. Show that $$ Y_{1}=n X_{(1)}, \quad Y_{r}=(n+1-r)\left(x_{(r)]-X_{(r- 1)} ight), \quad r=2, \ldots, n $$ are also independent and have the same joint distribution as $X_{1}, \ldots, X_{n}$. Hint: You may use the fact that the determinant of a lower triangular matrix (a square matrix whose entries above the main diagonal are all zero) is the product of the diagonal entries. S.P.PB. 328