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Poisson probability

  1. Jun 1, 2014 #1
    Suppose that a system is such that in a time dt, the probability that an event A occurs, given that it has not already happened, is given by:
    P(t,t+dt) = w(t) * dt

    The solution for the probability that A has occured at a time t is something like:

    P(t) = 1 - exp(∫0tw('t)dt')

    Now suppose that w(t) is changing due to some function x such that really w(t) = w(x(t)). How do I find the probability P(x) that the event A has occured as a function of x?
     
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  3. Jun 1, 2014 #2

    HallsofIvy

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    The chain rule (or substitution for integrals):
    [tex]\int w(t')dt'= \int w(x(t'))\left(\frac{dt'}{dx}\right)\left(\frac{dx}{dt'}\right) dt'= \int w(x)\left(\frac{dt'}{dx}\right)dx= \int_{x(0)}^{x(t)} \frac{w(x)}{\frac{dx}{dt'}}dx[/tex]
     
  4. Jun 1, 2014 #3
    But doesn't that still give me P(t) when I carry out the integration? My problem is that I have 3 different x1(t), x2(t), x3(t) and I want P(xi) for each of these so I can see when the probability in each case becomes significant. So in the above formula when I plug in x(t) in the end I still get something which is controlled by t. Where am I confusing myself?
    I also find it weird that if w(x) is an increasing function then the probability P(x) takes longer to reach a significant value when dx/dt is large. Intuitively I would think that if x(t) is driven to large values faster such that w(x) gets larger faster then the probability P(x) should do the same. Where am I wrong?
     
    Last edited: Jun 1, 2014
  5. Jun 1, 2014 #4

    Ray Vickson

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    As I read it, you have three "curves ##x_1(t), x_2(t), x_3(t)## and for each of them you have a (non-homogeneous) Poisson process with rate ##r_i(t) = w[x_i(t)]##. That will give you three different "counting processes" ##N_1(t), N_2(t), N_3(t)## with
    [tex] P\{ N_i(t) = n \} = \frac{m_i(t)^n \, e^{-m_i(t)}}{n!}, \, n = 0,1,2,\ldots [/tex]
    and where
    [tex] m_i(t) = \int_0^t w[x_i(\tau)] \, d \tau , i = 1,2,3. [/tex]
    That seems to be what you are saying; if it is not, you need to clarify what you want.
     
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