5. (10 points) The lifetime of a printer costing $500 is exponentially distribut

Question

5. (10 points) The lifetime of a printer costing $500 is exponentially distributed with a mean of 3 years. The manufacturer agrees to pay a full refund to the buyer if the printer fails during the finst year following its purchase, a one-half refund if it fails during the second year and a one-quarter refund during the third year. (a) (3 points) If the manufacturer sells 100 printers, how much is it expected to pay in re- funds the first year? (b) (3 points) If the manufacturer sells 100 printers, how much is it expected to pay in re- funds the second year? (c) (4 points) What is the expected total amount it pays in refunds for the 100 printers?

Explanation / Answer

The following information is provided:

The provided mean is ?=3.

Answer a)

We need to compute Pr(X?1).

Therefore, the following is obtained:

Pr(X<1) = 1 - e-1/3 = 0.2835

So that the expected number of failures in the first year out of 100 printers is 28.35 (100*0.2835)

Manufacturer is expected to pay = 28.35*500 = $14175

Answer b)

We need to compute Pr(1?X?2). Therefore, the following is obtained:

Pr(1?X?2) = Pr(X?2) ? Pr(X?1) = e-1/3 - e-2/3 = 0.7165 - 0.5134 = 0.2031

So that the expected number of failures in the second year out of 100 printers is 20.31 (100*0.2031)

Manufacturer is expected to pay = 20.31*(500/2) = $5077.5

Answer c)

We need to compute Pr(2?X?3). Therefore, the following is obtained:

Pr(2?X?3) = Pr(X?3) ? Pr(X?2) = e-2/3 - e-3/3 = 0.5134?0.3679 = 0.1455

So that the expected number of failures in the third year out of 100 printers is 14.55 (100*0.1455)

Manufacturer is expected to pay = 14.55*(500/4) = 1818.75

Expected total value manufacturer pays for 100 printers = Sum of expected Amount paid for year 1,2, and 3

Expected total value manufacturer pays for 100 printers = 14175 + 5077.5 + 1818.75 = $21071.25

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