# Question: Let X i i 1 n

Let X(i), i = 1, . . . , n, denote the order statistics from a set of n uniform (0, 1) random variables, and note that the density function of X(i) is given by

f(x) = n!/(i − 1)!(n − i)! xi−1(1 − x)n−i 0 < x < 1

(a) Compute Var(X(i)), i = 1, . . . , n.

(b) Which value of i minimizes, and which value maximizes, Var(X(i))?

f(x) = n!/(i − 1)!(n − i)! xi−1(1 − x)n−i 0 < x < 1

(a) Compute Var(X(i)), i = 1, . . . , n.

(b) Which value of i minimizes, and which value maximizes, Var(X(i))?

**View Solution:**## Answer to relevant Questions

Show that Y = a + bX, then Consider Example 4f, which is concerned with the multinomial distribution. Use conditional expectation to compute E[NiNj], and then use this to verify the formula for Cov(Ni,Nj) given in Example 4f. Show that, for random variables X and Z, E[(X − Y)2] = E[X2] − E[Y2] where Y = E[X|Z] Let X be a nonnegative random variable. Prove that E[X] ≤ (E[X2])1/2 ≤ (E[X3])1/3 ≤ . . . If X is a Poisson random variable with mean λ, show that for i < λ, P{X ≤ i} ≤ e−λ(eλ)i / iiPost your question