rphw5solf - Stochastic Processes nctuee07f Homework 5...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Stochastic Processes nctuee07f Homework 5 Solutions 1. Suppose the number of fish that a fisherman catches follows a Poisson process with rate λ = 0 . 5 per hour. The fisherman will keep fishing for two hours. If he has caught at least one fish, he quits. Otherwise, he continues until he catches at least one fish. Let N t denote the Poisson process of the problem. (a) This is the probability of no arrivals in 2 hours. It is given by P ( N 2 = 0) = e - 0 . 5 · 2 = e - 1 . (b) If he catches at least three fish, he must have fished for exactly two hours. Hence, the desired probability is equal to the probability that the number of fish caught in the first two hours is at least three, P ( N 2 3) = 1 - P ( N 2 = 0) - P ( N 2 = 1) - P ( N 2 = 2) = 1 - e - 1 - e - 1 - 1 2 e - 1 = 1 - 5 2 e - 1 . (c) Find the expected number of fish that he catches. Let N be the number of fish the fisherman catches. Let A denote the event that he catches at least one fish in 2 hours. Then, E [ N ] = E [ N | A ] P ( A ) + E [ N | A c ] P ( A c ) = (0 . 5 · 2)(1 - e - 1 ) + 1 · e - 1 = 1 .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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