Hint: First find the distribution for the total claim number. If you have four independent Poisson distributions, then what is the distribution of the sum?
Suppose that Ni (t) is a Poisson process with intensity λi, and you model
the number of insurance claims by the Poisson process for four dierent regions
with intensities λ1=20, λ2=25, λ3=35, and λ4 = 55 per month, respectively,
find the probability that the number of claim per year for those four regions are
larger than 1350.