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Exercise for December 29, 2007
May 2001 Course 1 Examination, Problem No. 35, also Study Note P-09-07, Problem
No. 63
The warranty on a machine specifies that it will be replaced at failure or age 4, whichever
occurs first. The machine’s age at failure, X, has density function
!1
# , for 0 < x < 5.
fX ( x ) = " 5
#$ 0, otherwise.
Let Y be the age of the machine at the time of replacement. Determine the variance of Y.
A. 1.3
B. 1.4
C. 1.7
D. 2.1
E. 7.5
Solution.
The random variable Y is effectively defined as the following function of:
" X, if 0 ! X ! 4,
Y = h( X ) = #
$ 4, if 4 < X ! 5.
Therefore,
5
4
5
1
1
1 2
E (Y ) = E ( h ( X )) = ! h ( x ) f X ( x ) dx = ! x " dx + ! 4 " dx =
x
5
5
10
0
0
4
( )
(
)
4
5
1
1
1 3
2
E Y 2 = E ( h ( X )) = " x 2 ! dx + " 4 2 ! dx =
x
5
5
15
0
4
and
( ) ! ( E (Y ))
Var (Y ) = E Y
2
2
2
112 " 12 %
=
! $ ' ( 1.71.
15 # 5 &
Answer C.
© Copyright 2004-2007 by Krzysztof Ostaszewski.
x=4
+
x=0
x=4
+
x=0
4 16 4 12
=
+ = ,
5 10 5 5
16 64 16 112
=
+
=
,
5 15 5
15
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