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Poisson process question

  1. Mar 2, 2010 #1
    I'm a bit frustrated with this one...

    Let [itex](X_t)_{t\geq 0}[/itex] be a Poisson Process with rate [itex]\lambda[/itex]

    Each time an 'arrival' happens, a counter detects the arrival with probability [itex]p[/itex] and misses it with probability [itex]1-p[/itex]. What is the distribution of time, [itex]T[/itex] until the first particle is detected?

    I thought I could do it like this:

    [tex]P(T = t) = \sum_{n=1}^\infty P(T = t | X_t = n) P(X_t = n) = \sum_{n=1}^\infty (1-p)^{n-1}p \frac{e^{-\lambda t}(\lambda t)^n}{n!}[/tex]

    Unfortunately, this does give me the right answer (T should be Exponentially distributed with parameter [itex]\lambda p[/itex]). I've stared at my process, but I can't seem to find anything wrong with it!

    Can someone point out the error in my approach?
  2. jcsd
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