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u 11 Pr{1 < A < 2 + dl, k eventsin 0 to t} fp(1|(k't))d2 = { <Pr<k events in 0 tot}-}

_ Pr{k events in 0 to t|l}f (l)dl Pr{ k events in 0 to t}

The key here is the transition from the numerator in the upper expression to the one in the lower expression, which relies on the definition of conditional probability, Pr{A, B} = Pr{A|B}Pr{B}, and recalling thatf(2)d2 = Pr{l < L < 2 + dl}. Now, we know each of the probabilities that are called for on the right hand side of Eq. (3.67): the probability of k events given 2 is Poisson, the probability density for l is gamma, and the probability of k events is negative binomial. Substituting the appropriate distributions, we have

Although this equation looks to be a bit of a mess, it actually simplifies very nicely and easily to become fp(2 I k) = + ! , , e-(a+'>22v+k-1 (3.69)

We recognize this as another gamma density, with changed parameters: we started with parameters a and v, collected data of k events in 0 to t, and update the parameters to a +1 and v + k, while keeping the same distribution forthe encounter rate. In the Bayesian literature, we say that the gamma density is the conjugate prior for the Poisson process (see Connections).

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