P_PracticeExam6_05-1..

PRACTICE EXAMINATION NO. 6
21. Let X( 1)
,..., X( n)
be the order statistics from the uniform distribution on [0, 1]. Find the
correlation coefficient of X( 1)
and X( n)
.
A. ! 1
n
B. ! 1
n + 1
C. 0 D. 1
n + 1
1
E.
n
Solution.
If X is uniform on [0, 1], so is 1 – X, and order statistics for a random sample for X are in reverse
order of the order statistics of 1 – X, so that
( ) = 1 ! E X n
E X( 1)
Var X( 1)
( ( ) ),
( ) = Var( 1! X( n)
) = Var X n
Observe that
( x)
= 1! x n ,
sX( n)
so that
E( X( n)
) = 1 ! x n
1
" ( ) dx =
0
n
n + 1 .
This implies that
( ) = 1 ! E X n
E X( 1)
We also have
x
and
fX( n)
2
E X n
This implies that:
( ) = nx n!1 ,
( ( ) ) = x 2 ! nx n"1
1
#
0
( ) = n
( ( ) ) = 1! n
n + 1
1
( ( ) ).
= 1
n + 1 .
dx = n x n+1
# dx = n
n + 2 .
0
( )2 ! n 2 n + 2
( )
( ) 2 ( n + 2)
( ( ) ).
Var X( n)
n + 2 !
n 2
n n + 1
n
2 = =
( n + 1)
n + 1
( n + 1)
2 ( n + 2)
= Var X 1
The joint density of X( 1)
and X( n)
is determined from observing that if X( 1)
is in (x, x + dx) and
X( n)
is in (y, y + dy), with x < y, then n – 2 pieces of the random sample must be in the interval
( x, y),
and the probability of this is
n!
fX( 1)
, X ( x, y)
= y ! x
( n)
( n ! 2)!
( )n!2 dxdy.
Therefore,
ASM Study Manual for Course P/1 Actuarial Examination. © Copyright 2004-2007 by Krzysztof Ostaszewski - 277 -