Question

A machine has three components that are identical and they operate independently of each other. The machine will operate as long as any one of three of the components is operating, The lifetime of each component follows an exponential distribution with a mean lifetime of 2 years. Determine the probability the machine is still operating after 1.5 years. Suppose the machine in the previous problem requires all three of the components to operate. If the lifetime of each component follows an exponential distribution with a mean lifetime of 2 years. Determine the probability the machine is still operating after 1.5 years.

Answer 1

solution: Con sider a machine has three compo ne nts the are identical and operate independently of each other The machine wri operate as long cus omy one of the three components is opesatina 1his means thait the three companents are arranged paralley es follows :「 c machine components os a 3 machine Also given helt ufe time of each component loDS on exponential clistaibution uith a mean lifetime of 2 yearS cdf of exponenticl clist oibution is, l, We know thet , in parauel care PC system wrt survivet)FtFct) 3 3 七 e 2 Here t s years ) F(t) = 1- (I-e . a -CI-0.4224)1-(0.5276) ご1-0. I468 Fct) 30.8532 So the pro bability thet the machi ne is still opesatina after Is years is .8S 32.

Now, given thel machine opexcites if all the Ehree compone r/ s are o peyalima This means thot the components are arra nge series 2 We knouw that, Tn seaies case P ( system will ramrtue > t) = F(t) 讼, esty 2 3 r(1-o-472の C.5276 So the Poba b1Utg that the machine s still operating after 1.s years s o.14 68