Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 2 / 3F (Probability)

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- Solution 3F

Solution

For

0 \leq u \leq 1

, independence gives

P (U > u) = P (X > u, Y > u) = (1 - u)^{2} .

Thus

U

has probability density function

f_{U} (u) = 2 (1 - u)

, and its expected value is

E U = \int_{0}^{1} 2 u (1 - u) d u = \frac{1}{3} .

U + V = X + Y

and

U V = X Y

. By linearity of expectation,

E V = 1 - 1/3 = 2/3

; by independence,

E (U V) = E X E Y = 1/4

. Consequently the covariance of two uniform order statistics is

Cov (U, V) = E (U V) - E U E V = \frac{1}{4} - \frac{2}{9} = \frac{1}{36} .

The order statistics are dependent despite the independence of the original observations.

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