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    Right. X_1 ~ Poisson(A), X_2 ~Poisson(B). They are independent. We are interested in the distribution of A given that their sum S = A + B is some particular value, s, say.

    Right. Well,
    P(X_1=x | S=s) = \frac{P(X_1=x)P(X_2=s-x)}{P(S=s)} = \displaystyle \binom{s}{x} (A+B)^{-s} A^x B^{s-x}, and now I'm supposed to be able to name this distribution and hence state its expected value. I'm certain I've worked this out correctly but I can't work out what distribution this is! It's discrete and there's only so many it can be! Help please!

    EDIT: Never mind I worked it out, it's distributed Binomial(s,\frac{A}{A+B}).
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