# Question

Let X(1) ≤ X(2) ≤ . . . ≤ X(n) be the ordered values of n independent uniform (0, 1) random variables.

Prove that for 1 ≤ k ≤ n + 1,

P{X(k) − X(k−1) > t} = (1 − t)n

where X(0) ≡ 0, X(n+1) ≡ t.

Prove that for 1 ≤ k ≤ n + 1,

P{X(k) − X(k−1) > t} = (1 − t)n

where X(0) ≡ 0, X(n+1) ≡ t.

## Answer to relevant Questions

Let X1, . . . ,Xn be a set of independent and identically distributed continuous random variables having distribution function F, and let X(i), i = 1, . . . , n denote their ordered values. If X, independent of the Xi, i = ...Let X and Y be independent continuous random variables with respective hazard rate functions λX(t) and λY(t), and set W = min(X,Y). (a) Determine the distribution function of W in terms of those of X and Y. (b) Show that ...Let X have moment generating function M(t), and define ψ(t) = logM(t). Show that ψ′′(t)|t=0 = Var(X) A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card. A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in ...Post your question

0