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Poisson Process Conditional Distribution

  1. Apr 4, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]X_t[/itex] and [itex]Y_t[/itex] are poisson processes with rates [itex]a[/itex] and [itex]b[/itex]

    [itex]n = 1,2,3...[/itex]


    Find the CDF [itex]F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)[/itex]


    2. Relevant equations



    3. The attempt at a solution
    [itex]F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)[/itex]

    [itex]=P(X_t<x|X_t+Y_t=n)[/itex]

    [itex]=\frac{P(X_t<x,X_t+Y_t=n)}{P(X_t+Y_t=n)}[/itex]

    Not sure from here, but here goes:

    [itex]=\frac{P(Y_t>n-x)}{P(X_t+Y_t=n)}[/itex]

    [itex]=1-\frac{P(Y_t<=n-x)}{P(X_t+Y_t=n)}[/itex]



    Not sure if doing correctly.
     
  2. jcsd
  3. Apr 4, 2013 #2

    Ray Vickson

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    Since X and Y are counting processes, you should probably avoid using the letter 'x' for values of them, so instead, should use something like ##F_{X_t|X_t + Y_t = n}(m).## Note also that the standard definition of a cdf involves '≤', not '<', so
    [tex]F_{X_t|X_t + Y_t = n}(m) = P(X_t \leq m|X_t+Y_t = n).[/tex]
     
  4. Apr 4, 2013 #3
    Can someone please help me with my solution, whether I am on the right track in my steps to get to a solution. Thanks
     
  5. Apr 5, 2013 #4

    Ray Vickson

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    All you have done is use the definition of conditional probability; you are nowhere near the final solution.
     
  6. Apr 5, 2013 #5
    ok,thanks
     
    Last edited: Apr 6, 2013
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