Question

2 (20 polats) A lighting system is comprised of two lightbulbs work indepeedently. Manufacturer fested failure of these components and this failure is know n to oceur randomly with rate (.) of 0.5 per year (a) Define the lifetime random variable and its pdf function. What is the expected lifetime? What is the 80% (i.e., top 20%) lifetime years? (b) What is the probability that both lightbulbs are still functioning after 2 years? (Hint: calculate probability of one component functioning after 2 years first)

Answer 1

a) here lifetime random variable X follows exponential distribution with paramter $\small \lambda$ =0.5

expected life time =1/ $\small \lambda$ =1/0.5 =2 years

80 th percentile life time = - ln(1-p)/ $\small \lambda$ =-ln(1-0.8)/0.5 =3.22 Years

b)

probability that one light bulb last more than 2 years =P(X>2) =e^{- $\small \lambda$ x} =e^-0.5*2 =0.3679

hence probability that both light bulb last more than 2 years =0.3679*0.3679 =0.1353