An automobile insurance company classifies each of its policyholders as being of one of the types i = 1, . . . , k. It supposes that the numbers of accidents that a type i policyholder has in successive years are independent Poisson random variables with mean λi, i = 1, . . . , k. The probability that a newly insured policyholder is type i is pi,

ki

=1 pi = 1. Given that a policyholder had n accidents in her first year, what is the expected number that she has in her second year? What is the conditional probability that she has m accidents in her second year?