Solution:-
Given that
A parallel system consists of two components with independent lifetimes
. The lifetime of L1 of the first component is memoryless; it has the exponential distribution with parameter .
Based on statistical analysis, it is found out that the lifetime L2 of second component has two independent phases each of which has the same characterisitics as L1. Therefore, it is assumed that L2 has gamma distribution with shape index K = 2 and scale parameter .
What is the expected lifetime of this system?
pdf for life time :
pdf for life time , which is gamma distribution with shape index k = 2 and scale parameter .
where
at k = 2
therefore pdf is
As and are independent then the sum of and is also independent and their pdf is also.
pdf of the system is
Expected life time for system
After solving
we get
Therefore,
Expected lifetime for system,
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