In the long run what fraction of the street is

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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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