Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 26J / c

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- Solution c

Solution

The three stage counts are independent Poisson variables of mean

ρ

under the product law. Hence

P (N = n) = e^{- 3 ρ} \frac{( 3 ρ ) ^{n}}{n !} .

Dividing the joint law by this mass gives

P (C = c, R = r, B = b ∣ N = n) = \frac{n !}{c ! r ! b !} (\frac{1}{3})^{n}, c + r + b = n .

Thus the conditional stage composition is multinomial with equal stage probabilities.

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