Probability And Statistics | Week 4
Probability And Statistics Week 4 Answers
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Q1. The number of phone calls over a certain time period follows a Poisson distribution. What is the average number of calls necessary for the probability of one or more calls is at least 0.98?
Q2. The total duration of all games by one team in one season of a major baseball league is seen to range uniformly between 427 hours to 504 hours. What is the probability that a team will play between 479 to 498 hours of games in a season?
Q3. Assume that the time taken to finish a standard sized piece of cake by a nine-year-old child is uniforml distributed between 0.5 and 3 minutes. Given that a child has been eating the cake for more than 1.5 minutes, determine the probability that he/she will finish it in more than two minutes?
Q4. Maximum time to complete a project is 2.5 days. Suppose the completion time as a proportion of this maximum time is a beta random variable with a =2 and ß = 3. What is the probability that the project requires more than two days to complete?
Q5. With an average of 10,000 miles, the number of miles a particular car may travel before experiencing battery failure is exponentially distributed. The car’s owner needs to travel 4000 miles. What is the probability that the journey can be finished without a car battery replacement?
Q6. The life of an MRI machine is modelled by a Weibull distribution with cumulative density function given by F(x) = 1 – e-(-², x ≥ 0,
where parameters ß = 3 and 8 = 700 hours. What is the probability that the MRI fails before 350 hours?
Q7. The time needed to solve a problem has an exponential distribution with Λ = 1/3. Two students, start tackling the problem simultaneously. (Assume the students are working independently on their own.) At the end of the two minutes, what is the probability that at least one student has figured out the problem?
Q8. Suppose X is a random variable with distribution function to be a Beta distribution (Beta(a, ß)). The mear and standard deviation of X are 0.09 and 0.03 respectively. Find the parameters (a, ß).
(A) (a, ß) = (8.1,81.9)
(B) (a, ß) = (3.8,91.2)
(C) (a, ß) = (7.1,82.9)
(D) (a, ß) = (17.1,72.9)
Q9. A Poisson process governs website hits, with a mean of 20 every hour. What is the probability that there won’t be any hits on the website in a 12-minute period?
Q10. Calls to a customer care number follow a Poisson distribution with a mean of 25 calls per minute. What is the mean time until the 100th call?
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