Let X (1) ≤ ··· ≤ X (n) denote the order statistics from a random sample of size n from an exponential distribution with rate λ. Consider the distances between points D 1 = X (1) , and D k = X (k) −X...

Let X₍₁₎
≤ ··· ≤ X_(n)
denote the order statistics from a random sample of size n from an exponential distribution with rate λ. Consider the distances between points D₁
= X₍₁₎, and D_k
= X_(k)
−X_(k−1), 2 ≤ k ≤ n. Show that these distances are independent, and that Dk has an exponential distribution with rate (n − k + 1)λ.