Page 250 - DMTH404_STATISTICS

Statistics

Notes Let P (t) be the probability that component i is functioning at time t. Then
i
P (t) P{component i is functioning at time t}

i
 P{lifetime of i  t}
 1 F (t)

i
 F (t).
i
Now let F be the distribution function for the lifetime of the system. How does F relate to the F ?
i
Let r(p) be the reliability function for the system, then


F(t) 1 F(t)
 P{lifetime of system  t}
 P{system is functioning at time t}

 r(P (t), .P (t))
n
1
 r(F (t), ,F (t)).
1 n
Example 1: The Series System

n
r( )   p ,
p
i
i 1

so that
n
F(t)   F (t).
i
i 1

Example 2: The Parallel System

n
p

r( ) 1  (1 p ),

i
i 1

so that
n


F(t) 1   (1 F (t)
i
i 1

n
 1   F . i
i 1

Failture Rate
 For a continuous distribution F with density f, the failure (or hazard) rate function of F, l(t), is
given by
f(t)
 (t)  .
F(t)
 If the lifetime of a component has distribution function F, then l(t) is the conditional
probability that the component of age t will fail.

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