2. Given that the lifetime of a particular device follows an exponential distribution with a mean of x hours, find the expected lifetime of the device, given that it has lasted at least Y hours. (15%)

3. A rare medical condition affects only 1 in 50,000 people in a specific population. A sophisticated diagnostic test for this condition has the following characteristics:●If a person has the disease, the test is \( 99.5\% \) accurate in detecting it (sensitivity).●If a person does not have the disease, the test has a \( 1.5\% \) false positive rate.●Two independent tests are performed on the same patient.What is the probability (%) that a person actually has the disease, given that they tested positive on the first test and also tested positive on the second test? (20%)(Note: You are asked to answer in percentages, counted to two decimal places.)

2. Given that the lifetime of a particular device follows an exponential distribution with a mean of \( x \) hours, find the expected lifetime of the device, given that it has lasted at least \( Y \) hours. (15%)

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