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

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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