Let Nt and Mt be independent Poisson processes with parameters lambda and mu, re

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Question

Let Nt and Mt be independent Poisson processes with parameters lambda and mu, respectively. Show P(Nt = k|Nt + Mt = n) = n!/k!(n - k)!(lambda/lambda + mu)k(1 - lambda/lambda + mu)n-k, that is the binomial with parameters n and lambda/(lambda + mu). Suppose that at a certain region of California, earthquakes occur at a Poisson rate of 8 per year. On the Richter scale, earthquakes of magnitude 5.5 or higher represent 2%. Suppose that last month, exactly 2 earthquakes occurred at this region, what is the probability that at least one of them has magnitude 5.5 or higher? (Hint use the formula from a.)

Explanation / Answer

P(Nt=k,Ml=n-k)/P(Nt+Ml=n)=e^-k*lambda^k/k!*e^-(n-k)miu^(n-k)/(n-k)!/e^-n*(lambda+miu)^n/n!=the rhs

as Nt+Ml follows poisson lambda + miu

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Let Nt and Mt be independent Poisson processes with parameters lambda and mu, re

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