Question

Imagine an electrical circuit with three components wired in series. This means that if any of the components fails, then the circuit will fail. Suppose that each component has an exponentially distributed lifetime, with ?1=0.235,?2=0.129, and ?3=0.274 (units measured in years). Give your answers to three decimal places.

What's the chance the circuit is still working after one year?  

The circuit ended up working for the first year. What's the chance it will last another two years?  

What's the expected time until the circuit fails?  years.

Answer 1

1) P(circuit is still working after one year)=P(1st component not fail)*P(seconf not fail)*P(third not fail)

=(e^-0.235*1)*(e^-0.129*1)*(e^-0.274*1)=0.5283

2)

as exponential distribution is memoryless; e chance it will last another two years

=(e^-0.235*2)*(e^-0.129*2)*(e^-0.274*2)=0.2792

3)

here probability that  circuit fails in time t =P(X<x)=1-P(none of comonent fails in time x)

=1-(e^-0.235*x)*(e^-0.129*x)*(e^-0.274*x)=1-e^-0.638x

which is exponential distribution with paramter =0.638

hence expected time until the circuit fails=1/ =1/0.638=1.5674 Years