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5. An electronic switching module is expected to malfunction some of the time, misdirecting messages. It must be replaced if the rate of such errors becomes too high. Letting  represent the mean rate of misdirected messages, the switch is operating according to specifications when  =0.10 per hour. Should  reach 0.50 error per hour, the module ought to be replaced
There is no way to know  precisely. A monitoring device may record the number of misdirection in a 10-hour test. Policy is to replace any module that causes more than 2 errors in the test, and otherwise to leave it in place.
a) What is the probability that a module needing replacement is retained?
b) What is the probability that a module is replaced even though it is operating according to specifications?

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Let X denote the time to failure (in year) of a certain hydraulic component. Suppose the pdf of X is
F(x) =32/(x+4) 3 for x>0.
a. Verify that f(x) is legitimate pdf.
b. Determine the cdf.
c. Use result of part (b) to calculate the probability that time to failure is between 2 and 5 years.
d. What is the expected time to failure?
e. If the component has a salvage value equal to 100/(4+x) when its time to failure is x , what is the expected salvage value?

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