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miyah5316
17.02.2021 •
Mathematics
A company estimates that 0.5% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $200. If they offer a 2 year extended warranty for $11, what is the company's expected value of each warranty sold?
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Ответ:
The company is expected to make
on every extended warranty sold.
Step-by-step explanation:
Let
denote whether a product with the extended warranty requires replacement in that two years. (
would be a random variable.) Assume that
means that the product requires replacement, and
otherwise.
Assume that this product requires replacement in that two years. That is:
. The company would then bear a cost of
for replacing this product (since
, that cost would be the same as
.)
On the other hand, assume
(that is: this product does not require replacement in that two years.) The company would not need to pay for replacing this product. Since
, the expression
would still represent the cost for the company for this warranty.
Either way,
would denote the replacement cost that the company would bear for this product. However, given that the company would charge
for the extended warranty, the net revenue of the company on this warranty would be
. (An earning of
minus a spending of
.)
The question states that
of the products would need replacement in this period. In other words, the expected value of
would be
.
The expected revenue of the company on this warranty would be:
Apply the linearity of expected values to find the value of
:
Hence, the company is expected to make
on every extended warranty that it sold.
Ответ: