Question

Suppose that a system has a component whose time in years until failure is nicely modeled by an exponential distribution. Assume there is a 60% chance that the component will not fail for 5 years.

a) Find the expected time until failure.

b) Find the probability that the component will fail within 10 years.

c) What is the probability that there will be 2 to 5 failures of the components in 10 years.

Answer 1

a)

let expected time untill failure is

therefore P(will not fail for 5 years) =P(X>5) =e^-x/ =e^-5/ =0.60

=-5/ln(0.60)=9.788 years

b)

probability that the component will fail within 10 years =P(X<10)=1-exp(-10/9.788)=0.64

c)

expected number of failures in 10 years =1/9.788 =1.022

therefore from Poisson distribution:

probability that there will be 2 to 5 failures of the components in 10 years

=e^-1.022*1.022²/2!+e^-1.022*1.022³/3!+e^-1.022*1.022⁴/4!+e^-1.022*1.022⁵/5!

=0.2715