a.
Given, P(Ti 100) = 0.99 for i = 1,2,...,10
Probability that the system will not fail in 100 hrs. = P(T 100) = P(min(T1, T2,..., T10) 100)
= P(T1 100, T2 100, ......, T10 100)
= P(T1 100) * P(T2 100) * .... * P(T10 100)
= 0.99 * 0.99 * ... * 0.99
= 0.9910
= 0.9043821
b.
Given, P(Ti 100) = 0.99 for i = 1,2,...,10
=> P(Ti 100) = 1 - 0.99 = 0.01
Probability that the system will not fail in 100 hrs. = P(T 100) = P(max(T1, T2,..., T10) 100)
= 1 - P(max(T1, T2,..., T10) 100)
= 1 - P(T1 100, T2 100, ......, T10 100)
= 1 - P(T1 100) * P(T2 100) * .... * P(T10 100)
= 1 - 0.01 * 0.01 * ... * 0.01
= 1 - 0.0110
= 1
c.
Given, Ti ~ Exponential() so, P(Ti t) = exp(-) for i = 1,2,...,n
Probability that the system will not fail in t hrs. = P(T t) = P(min(T1, T2,..., Tn) t)
= P(T1 t, T2 t, ......, Tn t)
= P(T1 t) * P(T2 t) * .... * P(n t)
= exp(-) * exp(-) * ... * exp(-)
= exp(-)
Thus, system failure time is exponentially distributed with parameter .
d.
Given, for all i = 1, 2, .., n
or,
Since, system failure time is exponentially distributed with parameter .
Expected component time until failure =
4. Reliability of Systems - Take n components to have failure times Ti, T2, ..., Tn...
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