I'm having some problem with this question.
A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the probability of another goal being scored in the next time interval (t, t+$\delta$t) is (v+$\lambda$n)$\delta$t +o($\delta$t), where the time scale is measured in minutes. Show that this leads to a negative binomial distribution for the numbers of goals per match. (Assume that the effective duration of a game is 90 minutes - any added time is supposed to make up exactly for stoppages due to injury or substitutions.)
I understand that this is a birth-immigration process and I can show that that the a pure birth process is a negative binomial distribution but I can't figure out how to incorporate the immigration process.Any help would be appreciated.