A 6 points what is the probability that exactly 3

Info icon This preview shows pages 3–5. Sign up to view the full content.

View Full Document Right Arrow Icon
people arrive randomly and independently. (a) [6 points] What is the probability that exactly 3 people arrive in the first 10 min- utes? Solution: We have a Poisson arrival process with rate λ = 7+5 = 12 per hour. So the number of people that arrive in the first 10 minutes is Poisson distributed with rate λ/ 6 = 2. Therefore, the probability of exactly three people is e - 2 2 3 3! 18 . 04% . (b) [6 points] If 3 people arrive in the first 10 minutes, what is the chance that 2 of them are women? Solution: Let A be the event “two women arrive in the first 10 minutes” and B be the event “three people arrive in the first 10 minutes.” So AB is the event “two women and one man arrive in the first 10 minutes”. Then P ( A | B ) = P ( AB ) P ( B ) = e - 5 6 (5 / 6) 2 2! e - 7 6 (7 / 6) 1 1! e - 2 2 3 3! = 175 576 30 . 38% . Note that the answer does not depend on the length of time. (c) [4 points] Let X be the number of men that arrive before the first woman and let Y be the arrival time of the first woman. What is E ( X | Y )? Page 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Math 431: Final Exam Solution: Given Y = y , the number of men X to arrive in (0 , y ) is Poisson dis- tributed with rate 7 y , hence it has mean E ( X | Y = y ) = 7 y . More compactly, E ( X | Y ) = 7 Y . (d) [4 points] Is Corr ( X, Y ) greater than, equal to, or less than zero? Explain. Solution: X and Y are positively correlated (i.e., Corr ( X, Y ) > 0) because the larger Y is, the larger we expect X to be. (e) [6 points] Find P ( X = 0). Solution: Given Y = y , as in part (c), X is Poisson distributed with rate 7 y .
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern