Question

1. The time to failure of a digital camera (in hours) is distributed exponentially with parameter 10^−4 .

a) Find the expected time to failure. (5)

b) Find the probability that the camera will last less than 9,000 hours or more than 12,000 hours. (10)

c) Find the probability that the camera will last more than 10,000 hours. (10)

d) If the camera has lasted 10,000 hours, find the probability that it will last another 10,000 hours or longer. (10 bonus)

Answer 1

here parameter λ =10^{-4 =}0.0001

F(x)=P(X<x)=1-e-λx

a)

expected time to failure =1/λ=1/0.0001=10000

b)

P(X<9000 or X>12000) =(1-e^-0.0001*9000)+(e^{-0.0001*12000})=0.8946

c)

P(X>10000)=e^{-0.0001*10000} =0.3679

d)

P(X>20000|X>10000) =P(X>20000)/P(X>10000) =e^{-0.0001*20000}/e^{-0.0001*10000} =e^{-0.0001*10000} =0.3679