Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Poisson Counting Process

  1. Oct 13, 2009 #1
    Let {N(t): t≥0} be a Poisson process of rate λ.
    We are given that for a fixed t, N(t)=n.
    Let Ti be the time of the ith event, i=1,2,...,n.

    Then the event {T1≤t1, T2≤t2,...,Tn≤tn, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t1], (t1,t2],..., (tn-1,tn], and no events occur in (tn,t].
    =====================================

    I don't understand the 'exactly one' part.
    For example, T2≤t2 just says that T2 is less than or equal to t2, and T2 can very possibly be less than t1 as well, right? (since it did NOT say that T2 MUST be larger than t1) In this case, we would then have more than one event occurring in [0,t1]. Why is this not allowed? I don't get it...

    Can someone please explain? I would really appreciate it!
     
    Last edited: Oct 13, 2009
  2. jcsd
  3. Oct 16, 2009 #2

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    Would N(t)=n be satisfied if T2 < t1?
     
  4. Oct 18, 2009 #3

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    It seems to be a definitional choice that serves some purpose in the rest of the problem.
     
  5. Oct 18, 2009 #4
    I think so! We can possibly have all n points in the interval [0,t1].
     
  6. Oct 18, 2009 #5
    The book is trying to prove that the joint density function of T1,T2,...,Tn given that N(t)=n is given by (n!)/(t^n), 0<t1<t2<...<t_n<t

    They are first trying to find the joint distribution function for 0<t1<t2<...<t_n<t.
    i.e. P(T1≤t1,...Tn≤tn |N(t)=n)
    and they commented that "the event {T1≤t1, T2≤t2,...,Tn≤tn, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t1], (t1,t2],..., (tn-1,tn], and no events occur in (tn,t". This is where I got totally confused...

    Can someone please help?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Poisson Counting Process
  1. Poisson Process (Replies: 6)

  2. Poisson Process (Replies: 4)

Loading...