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Independent Poisson Processes Word Problem

  1. Nov 14, 2012 #1
    Hello, hopefully this is the right place. This is a homework question, so it should definitely be in this forum, but I wasn't sure which sub-forum to put this rather elementary stats question.

    1. The problem statement, all variables and given/known data
    In my introductory mathematical statistics class, we've been given the following word problem having to do with Poisson processes.

    We are to consider a football game, where each team's score follows its own Poisson process. So there are two teams, team A and team B and, each team's final score has its own lambda parameter:

    Team A scores [itex]\lambda_A[/itex] goals per game.
    Team B scores [itex]\lambda_B[/itex] goals per game.

    What is the probability of team B winning over team A, with a final score of 3-2.

    2. Relevant equations
    3. The attempt at a solution
    My intuition (which I would like to confirm/have critiqued) is that since the processes are independent of each other I should find the following.

    Let [itex]X_A[/itex] be the final score of team A
    Let [itex]X_B[/itex] be the final score of team B

    [itex]
    Pr(X_A = 2) = \lambda_A ^ 2 e^{- \lambda_A } / 2!
    [/itex]
    [itex]
    Pr(X_B = 2) = \lambda_B ^ 3 e^{- \lambda_B } / 3!
    [/itex]

    And then the probability that team B wins with a score of 3 to 2 is simply the product of the two probabilities above, namely:

    [itex]
    Pr(X_A = 2 \cap X_B = 3) = Pr(X_A = 2) \cdot Pr(X_B = 3)
    [/itex]

    Is this the correct approach?
     
  2. jcsd
  3. Nov 14, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it is correct (given the rather unlikely scenario that the two scores are independent).

    RGV
     
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