Unformatted text preview: = e a· e z
1/
So in the limit the age and residual life are independent exponential.
Example 3.11. Uniform on (0,b). Plugging into (3.11) gives for a, z > 0,
a + z < b:
1/b
2
=2
b/2
b
The margin densities given in (3.9) are
(b x)/b
2⇣
= ·1
b/2
b x⌘
b In words, the limiting density is a linear function that starts at 2/b at 0 and
hits 0 at c = b.
Inspection paradox. Let L(t) = A(t) + Z (t) be the lifetime of the item in
use at time t. Using (3.10), we see that the average lifetime of the items in use
up to time t:
E (t2 )
1
> Eti
Et1
since var (ti ) = Et2 (Eti )2 > 0. This is a paradox because the average of the
i
lifetimes of the ﬁrst n items:
t1 + · · · + tn
! Eti
n
and hence t1 + · · · + tN (t)
! Eti
N (t) There is a simple explanation for this “paradox”: taking the average age of the
item in use up to time s is biased since items that last for time u are counted
u times. That is,
1
t Z 0 t A(s) + Z (s) ds ⇡ N (t)
N (t)
1X
1
·
· Et2
ti · ti !
1
t
N (t) i=1
Et1 113 3....
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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