Negative Binomial and Poisson Distributions Compared

For much statistical work the binomial distribution is the most suitable mathematical model. It involves n independent trials, each having a probability of success equal to p. In automobile and other branches of casualty insurance, we are not concerned with a limited number of independent trials, but with an exposure to accident such that n becomes very great while n X p remains finite and is the number of "successes". In this case, the Poisson distribution is the correct statistical model to produce the probability that 0, 1, 2 . . . . . "successes" will be experienced by a given observational unit (one car, a fleet of cars, all the cars in one territory, etc.).