A manufacturer uses exponential distribution to model the cy

A manufacturer uses exponential distribution to model the cycle-to-failure number of its products. In this case, r.v. T in the exponential pdf represents the number of cycles to failure lambda= 0.003 failure/cycle. a. What is the mean number of cycles to failure for this product? b. If a component survives for 300 cycles, what is the probability that it will fail sometime after 500 cycles? Accordingly, if 10(X) components have survived 300 cycles, how many would be expected to fail after 500 cycles?

Solution

(a) mean=1/0.003=333.3333

---------------------------------------------------------------------------------------------------

(b) F(x) =1-exp(-0.003*x) for x>0

So the probability is

P(X>500|X>300) = P(X>500) / P(X>300)

=(1-F(500))/ (1-F(300))

=(1-(1-exp(-0.003*500))) / (1-(1-exp(-0.003*300)))

=0.5488116

---------------------------------------------------------------------------------------------------

1000*0.5488116 = 548.8116