Question

In: Statistics and Probability

The time until failure of the hard drives on laptops currently sold in the United States...

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.

Solutions

Expert Solution

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


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