Poisson Approximation to the Binomial Distribution

Previously, we noted that the Poisson probability distribution is obtained by starting with the Binomial probability distribution with P approaching 0 and n becoming very large. Thus, it follows that the Poisson distribution can be used to approximate the binomial probabilities when the number of trials, n is large and at the same time the probability, P, is small (generally such that)

Example: An analysts predicted that 3.5% of all small corporations would file for bankruptcy in the coming year. For a random sample of 100 small corporations, estimate the probability that at least 3 will file for bankruptcy in the next year, assuming that the analyst’s prediction is correct.

Solution: The distribution of X, the number of fillings for bankruptcy, is binomial with n = 100 and P = 0.035, so that the mean of the distribution is . Using the Poisson distribution to approximate the probability of at least 3 bankruptcies, we find

The Poisson probability is simply an estimate of the actual binomial probability.