Exercise 4.15. Suppose that a class of students is star-gazing on top of the loc

Question

Exercise 4.15. Suppose that a class of students is star-gazing on top of the local mathematics building from the hours of 11 PM through 3 AM. Suppose further that meteors arrive (i.e. they are seen) according to a Poisson process with intensity ? -4 per hour. Find the following. our (b) The probability they see zero meteors in the first hour, but at least 10 meteors in the final three hours (midnight to 3 AM). (c) Given that there were 13 meteors seen all night, what is the probability there were no meteors seen in the first hour?

Explanation / Answer

a) P(x > 2) = 1 - P(0) - P(1) - P(2) = 1 - e^-4 - e^-4*4^1/1! - e^-4*4^2/2! = 1 - 13 e^-4 = 0.761896694446456

Alternatively, 1 - poisson.dist(2,4,TRUE) = 0.761896694446456, the same answer.

b) As lambda = 4 per hour, in 1 hour, P(0) = e^-4.

In 3 hours, the rate is 4*3 = 12. Thus, the probability of >= 10 events = 1 - P(0 - 9) with lambda = 12. We can either use the Poisson distribution to add P(0-9), then subtract from 1 or use Excel's 1 - poisson.dist(9, 12, True) = 0.757607838329488

Then, e^-4 * 0.757607838329488 = 0.0138760715861174

c) The probability that each of the meteors hits in the first of 4 hours is 1/4

Then, the probability that none of 13 strike is (1-1/4)^13 = (3/4)^13 = 0.0237572640180588

Related Questions

Navigate

• Browse (All)
• Browse E
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.