An insurance company has two insurance portfolios. Claims in Portfolio Poccur inaccordance with a Poisson process with mean 3 per year. Claims in portfolio Qoccur inaccordance with a Poisson process with mean 5 per year. The two processes areindependent.Calculate the probability that 3 claims occur in Portfolio Pbefore 3 claims occur inPortfolio Q. I proceeded with this as you received three claims out of five from p and the sixth claim will be from q, like a normal negative binomial. Solution however assumes as min 3 claims out of first five from p. I didn’t quite get the logic
Consider the process X = P + Q. This is also a Poisson process with mean 8 (=3+5). Probability of a claim in this process belonging to P is 3/8 and to Q is 5/8. If we think of a claim belonging to P as a success, then the random variable Y denoting the number of successes (claims from P) out of the first 5 claims is a Binomial random variable with success probability p = 3/8. Now we can ask what is the probability of Y=3,4 or 5.