In: Statistics and Probability
The time until failure of the hard drives on laptops currently sold in the United States follows an exponential distribution, with a mean of 4.2 years. A company offers a special incentive program that guarantees a $500 refund if the hard drive on their laptops fails in the first year after purchase, and a $100 refund if the hard drive fails in the second year after purchase. Find the expected refund a customer would receive after purchasing one of this company’s laptops.
Let T be the time until failure of the hard drives. Then T ~ Exp( = 1/4.2) ( = 1/mean)
Probability that the hard drive fails in the first year = P(T < 1) = 1 - exp(- * 1)
= 1 - exp(-1/4.2)
= 1 - exp(-0.2381)
= 0.2119
Probability that the hard drive fails between the first and second year = P(1 < T < 2) = P(T < 2) - P(T < 1)
= (1 - exp(- * 2)) - (1 - exp(- * 1)) = exp(- * 1) - exp(- * 2)
= exp(-1/4.2) - exp(-2/4.2)
= exp(-0.2381) - exp(-0.4762)
= 0.7881 - 0.6211
= 0.167
Expected refund a customer would receive after purchasing one of this company’s laptops
= $500 * P(T < 1) + $100 * P(1 < T < 2)
= 500 * 0.2119 + 100 * 0.167
= $122.65