?Does the random variable that has this distribution get its value from one event or one trial or from several?
?Does the random variable that has this distribution count the number of events (such as the number of successes) or the number of trials?
?Does the random variable that has this distribution get its value from one trial or several?
This table should guide your analysis of the study guide questions. This preparation will probably be important for the exam.
3.23 Every day, a lecture may be canceled due to inclement weather with probability 0.05. Class cancelations on different days are independent. (a) There are 15 classes left this semester. Compute the probability that at least 4 of them get canceled. (b) Compute the probability that the tenth class this semester is the third class that gets canceled.
3.25 About ten percent of users do not close Windows properly. Suppose that Windows is installed in a public library that is used by random people in a random order. (a) On the average, how many users of this computer do not close Windows properly before someone does close it properly ?(b) What is the probability that exactly 8 of the next 10 users will close Windows properly ?
3.27 Messages arrive at an electronic message center at random times, with an average of 9 messages per hour. (a) What is the probability of receiving at least five messages during the next hour? (b) What is the probability of receiving exactly five messages during the next hour ?
3.28 The number of received electronic messages has Poisson distribution with some parameter 入. Using Chebyshev inequality, show that the probability of receiving more than 4入messages does not exceed 1/(9入).
3.31 Before the computer is assembled, its vital component (motherboard) goes through a special inspection. Only 80% of components pass this inspection. (a) What is the probability that at least 18 of the next 20 components pass inspection ? (b) on the average, how many components should be inspected until a component that passes inspection is found ?
3.32 On the average, 1 computer in 800 crashes during a severe thunderstorm. A certain company had 4000 working computers when the area was hit by a severe thunderstorm. (a) Compute the probability that less than 10 computers crashed. (b)
Compute the probability that exactly 10 computers crashed.
3.36 An interactive system consists of ten terminals that are connected to the central computer. At any time, each terminal is ready to transmit a message with probability 0.7, independently of other terminals. Find the probability that exactly 6 terminals are ready to transmit at 8 o’clock.
3.37 Network breakdowns are unexpected rare events that occur every 3 weeks, on the average, Compute the probability of more than 4 breakdowns during a 21-week period.
In the course of stress testing 17,000 computer motherboards, AT&T determined that 50% of the defective boards showed intermittent failures. They failed only at high temperatures and then only sometimes. Over time, the failure rate of defective boards increases until eventually they have a hard failure (always fails). Thus a board that fails only rarely at first will gradually become non-functional. Suppose that a person buys an inexpensive laptop computer, and leaves it in his car the next day. The Florida sunshine cooks the computer, damaging a component. The computer fails the first time the customer tries to boot it, but then seems to work OK. Not trusting the machine, the customer takes it back to the store, says he doesn't like it and asks for a different machine. They let him exchange it and take back the computer. The store then runs diagnostics to determine whether the computer is fit for resale.
Suppose the store likes to use a "memory stress test", running it several times. Suppose too that at the store temperature, this computer will fail this test (on average) 1 time per thousand runs.
Question 9
Determine the probability that the computer will fail diagnostics if the store runs its test 15 times.
Question 10
(a) Suppose the store runs the diagnostic 100 times per night. Determine the failure rate parameter, lambda, for a night of testing. What is the probability that the computer will fail at least once in 1 night of testing?
(b) What is the probability the computer will fail at least once in 3 nights of testing?
(c) What is the failure rate parameter (for the number of failures) if the store runs 10 nights of tests on this computer?
Question 11
(a) What is the probability that the computer will not have its first failure until the 10th night of testing?
(b) Suppose the computer has its first failure on the first night of testing. Testing restarts the next night. What is the probability that the computer will have its next failure on the 11th night of testing?
Question 12
What is the probability that the 5th night on which there is a failure will be the 30th night of testing?
Question 13
How many nights will the store have to run tests on this computer if it wants a 95% chance of getting at least one failure?
Analyze this three ways and compare the results:
(a) Binomial distribution that counts the number of nights in which there is at least one failure.
(b) Poisson distribution that approximates the binomial distribution in (a)
(c) Poisson distribution that considers the probability of individual failures (100 tests per night)