Question

Suppose that the average lifetime of a car is 100 working hours and that the lifetime distribution is exponential.
(a) Estimate the probability that the car will work at least 50 hours.
(b) Given that the car has functioned for 50 hours, what is the chance that it fails in the next 25 hours.

(c) Suppose that two cars, one in active, the other in reserve (if the primary one malfunctions, the second will be used at once). What is the chance that these two cars remain working more than 50 hours?

Answer 1

average lifetime = 100

the distribution of exponential.

so here probability distribution function is

f(x) = (1/100) e-x/100 ; x > 0

cumulative probability distribution

F(x) = 1- e-x/100 ; x > 0

(a) f(x > 50) = 1 - F(50) = 1 - (1 - e-50/100) = 0.6065

(b) As we have to find here.

P(x > 50 + 25 l x > 50)

as we know that exponential distribution has memoryless property

P(x > 50 + 25 l x > 50) = P(x > 25)

= 1 - (1 - e-25/100) = exp(-0.25) = 0.7788

(c) Here for one car P(x > 50 hours) = 0.6065

P(Both will work) = 0.6065 * 0.6065 = 0.3679