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Homework Help: Poisson Process question

  1. Feb 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Events X, Y, Z are all Poisson processes. Event X has a rate of 1 per unit time , event Y has a rate of 2 per unit time and event Z has a rate of 3 per unit time.

    Find the probability that 2 events (of any type) occur during the interval (0, 3).

    2. Relevant equations

    Maybe this is relevant
    P(N=k) = [tex]\frac{(\lambda t)^k e^{-\lambda t}}{k!}[/tex]

    3. The attempt at a solution

    So [tex]\lambda_X[/tex] = 1, [tex]\lambda_Y[/tex] = 2 and [tex]\lambda_Z[/tex] = 4. Also k = 2 and t =3.

    Is it correct to think of it as say A = X [tex]\cup[/tex] Y [tex]\cup[/tex] Z. Then the answer is:

    P(N=2) = [tex]\frac{(7(3))^2 e^{-7(3)}}{2!}[/tex] where [tex]\lambda_A = 1 + 2 + 4 = 7. [/tex]
  2. jcsd
  3. Feb 24, 2010 #2
    I believe that this is correct. If X Y and Z are independent then a random variable say A=X+Y+Z would have a poisson distribution with rate of [tex]\lambda_X[/tex] +[tex]\lambda_Y[/tex]+[tex]\lambda_Z[/tex]
    Although you initially wrote [tex]\lambda_Z[/tex] =3 ,but put down 4.
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