Page 254 - DMTH404_STATISTICS

Statistics

Notes
Example 2: A parallel system with two independent components with different
exponential lifetime distributions is not IFR; in fact, (t) is initially strictly increasing, and then
strictly decreasing.

17.8 Expected System Life

Since system lifetime is non-negative, then


E[system life]   P{system life  t}dt
0

F
  r( (t))dt,
0
where


F (t) (F (t), ,F (t)).
1 n
Example 3: k-out-of-n System. If each component has the same distribution function
G, then

 n  n  i

E[system life]        G(t)  G(t)  n i dt.
i

0 i k  
Example 4: k-out-of-n System. Assume the expected component lifetime has mean .
Uniformly distributed component lifetimes:

i

2 n  n   t   t  n i
E[system life]       1     dt
i 

0 i k   2  2

2 (n k 1)


 .
n 1

Example 5: k-out-of-n System (uniformly distributed component lifetimes).
Parallel system (1-out-of-n):
2 n

E[system life]  .
n 1

Serial system (n-out-of-n):
2
E[system life]  .
n 1


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