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Math Help - Sum of two Poissons (not as easy as it looks!)

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    Sum of two Poissons (not as easy as it looks!)

    I can show that the sum of two Poisson processes of rates a and b is a Poisson process of rate a+b.

    Now, find the probability that the arrival of the combined (a+b) Poisson process comes from the process of rate a. I know that the answer must surely be a/(a+b), but how do we actually go about formally proving it?
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  2. #2
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    Quote Originally Posted by Amanda1990 View Post
    I can show that the sum of two Poisson processes of rates a and b is a Poisson process of rate a+b.

    Now, find the probability that the arrival of the combined (a+b) Poisson process comes from the process of rate a. I know that the answer must surely be a/(a+b), but how do we actually go about formally proving it?
    I'm not sure what exactly you mean by "the arrival" of a Poisson process, but I'm pretty sure what you need to prove eventually is that if X,Y are independent, such that the distribution of X is exponential with parameter a and the distribution of Y is exponential with parameter b, then P(X\leq Y)=\frac{a}{a+b}. I have seen this done on the forum a few times; for instance you can write, using independence, P(X\leq Y)=\int_0^\infty P(Y\geq x) a e^{-a x} dx= \int_0^\infty e^{-bx} a e^{-a x} dx, etc.
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