4-103. The lifetime of a mechanical assembly in a vibration test is exponentiall

Question

4-103. The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours a) What is the probability that an assembly on test fails in less than 100 hours? b) What is the probability that an assembly operates for more than 500 hours before failure? (c) If an assembly has been on test for 400 hours without a fail- ure, what is the probability of a failure in the next 100 hours? (d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently. (e) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume the assemblies fail independently

Explanation / Answer

Exponential distribution:
(a to b) (1/ß) (e^(-x/ß)) dx
where ß is the mean, in this case ß = 400
= (a to b) (1/400) (e^(-x/400)) dx
= (1/400) (a to b) (e^(-x/400)) dx

a)
a = 0, b = 100
= (1/400) (0 to 100) (e^(-x/400)) dx
= (1/400) (-400) [ e^(-x/400) ] (from 0 to 100)
= - [ e^(-1/4) - e^0 ]
= 0.2212

b)
a = 500, b =
= (1/400) (0 to ) (e^(-x/400)) dx
= (1/400) lim(c) (0 to c) (e^(-x/400)) dx
= (1/400) lim(c) [(-400) (e^-c - e^(-500/400)) ]
= -lim(c) (e^-c - e^(-5/4)) ]
= - (0 - e^(-5/4))
= e^(-5/4)
= 0.2865

c)
a = 400, b = 500
= (1/400) (0 to 100) (e^(-x/400)) dx
= (1/400) (-400) [ e^(-x/100) ] (from 400 to 500)
= - [ e^(-5/4) - e^(-4/4) ]
= 0.0814

a = 0, b = 400
= (1/400) (0 to 400) (e^(-x/400)) dx
= (1/400) (-400) [ e^(-x/400) ] (from 0 to 400)
= - [ e^-1 - e^0 ]
= 1 - 1/e
= 0.63212

0.0814/(1 - 0.63212) = 0.2213

d)
P(X 0)
= 1 - P(X = 0)
= 1 - (10C0)(0.2213^0)(1 - 0.2213)^10
= 0.9180

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