4. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter 0= 4. What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 1 hours?
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- Find the probability that a lot of 100 items, of which five are defective, will be accepted in a test of a randomly selected sample containing half the lot if, to be accepted, the number of defective items in a lot of 50 cannot exceed one.At a toll-booth on-ramp there is a stochastic arrival distribution. Vehicles are counted in 20-second intervals, and vehicle counts are taken in 120 of these time intervals. Based on data collected, no vehicles arrived in 18 of the 120 count intervals. What is the number of the 120 intervals that 3 cars arrived?2
- The lifetime (in years) of a washing machine from a particular company is an exponential random variable with parameter 1/5 . The company offers to replace any machine which breaks within 6 months. (a) What proportion of machines do they need to replace. (b) What is the expected time until a customer needs to buy a new machine? (Each time the company replaces a machine which breaks within 6 months, the company agrees to replace the new machine if it breaks within 6 months.)4. A package of 50 computer chips contains 45 that are perfect and 5 that are defective. If 2 chips are selected at random, what is the probability that a) Neither is defective? b) Both are defective? c) Only one is defective?The service time for a passport application in a certain government office is modelled as an exponential random variable with parameter λ = 5 per hour, independent and identically distributed for all applicants. Determine the probability that 50 applicants is accommodated within eight hours. Determine the maximum number of applicants such that the chance of accommodating all of them within eight hours is at least 95%.
- The expected life of a certain component in a mechanism is treated as a random variable having a gamma distribution with a = 3 and ß = 2. The average expected life of such component is 12 months, what is the probability that this component can have a greater than average expected life?A G's production of 950 manufactured parts contains 70 parts that do not meet the customer requirements . Two parts are selected randomly without replacement from the batch. What is the probability that the second part is defective given that the first part is defective?An average of 1 package is sent to a server in 1 second. The server, on the other hand, prepares a document using the 5 packages received. The time between packets is modeled as an exponential random variable. Let X be a random variable expressing the time it takes the server to prepare a document. The time taken to prepare a document; a) Find the probability that it is more than 10 seconds.(with detail) b) Find the probability that it is between 5 seconds and 10 seconds. (with detail)
- Find the probability that a lot of 100 items, of which five are defective, will be accepted in a test of a randomly selected sample containing half the lot if, to be accepted, the number of defective items in a lot of 50 cannot exceed one.If the lifetime of a computer is an exponential random variable whose mean is 5 year. Given that the computer has not broken down after 3 years, find the expected total lifetime of this computer. (a) 4 (b) 6 (c) 8 (d) 10