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Newton-Raphson method in non-homogeneous poisson process

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    The rate of occurrence of events in a non-homogeneous Poisson process is given by: λ(t)=12t e-2t.

    (c) Find the p.d.f. of the time until the first event occurs after time t = 1.
    (e) After what time is it 95% certain that no further events will occur?

    2. Relevant equations
    λ(t)=12t e-2t


    3. The attempt at a solution
    After using integration by parts, I found μ(t) = -3e-2t(2t+1) to solve other parts of this question. I know part (e) requires the use of newton-raphson method but I have no idea how to go about. Any help will really be appreciated, thanks.
     
    Last edited: Nov 12, 2011
  2. jcsd
  3. Nov 12, 2011 #2

    hotvette

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    Newton-Raphson is a numerical method to estimate the zero of a function (i.e. find x such that f(x) = 0) to some desired accuracy. Are you having difficulty defining f(x) or applying NR or both?
     
    Last edited: Nov 12, 2011
  4. Nov 14, 2011 #3
    Both.

    I know the probability of extinction by each generation can be calculated using Newton_Raphson method given p.g.f. and finding p and q, but I am not sure how to apply to this question.
     
  5. Nov 14, 2011 #4

    Ray Vickson

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    Are you trying to solve an equation? What IS the equation? Write it down in detail, so we have the basis for offering some advice.

    RGV
     
  6. Nov 16, 2011 #5
    I have managed to solve (c).

    Here is the equation i got for (e):
    g(t) = (3 + 6t)e^-2t - 0.0513 = 0
    g'(t) = -12te^(-2t)
    tn+1 = tn - [(3 + 6tn)e^-2tn - 0.0513]/-12tne^(-2tn)

    I am not sure how to proceed with NR to get the time of 95% certain?
     
  7. Nov 16, 2011 #6

    Ray Vickson

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    Why the concentration on Newton-Raphson? Do you understand that you are just trying to solve the equation (3 + 6t)*exp(-2t) = 0.051293? Newton-Raphson (NR) is one way to do it, but there are many others. However, if you do want to use NR to solve the equation g(t) = 0, you just start with some initial guess, t0, then use the iteration scheme
    tn+1 = tn - g(tn)/g'(tn). What is stopping you from doing this?

    RGV
     
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