As noted above, the Poisson distribution is obtained as an asymptotic limit of the binomial distribution when p is very small. The result is
where µ = np, as before, and 0! is understood to be 1. This distribution is determined by one rather than two constants: = (npq)^{1/2}, but q = 1 - p 1, so = (np)^{1/2} = µ^{1/2}. The standard deviation is equal to the square-root of the mean. The Poisson distribution is discrete: P(0; µ) = e^{-µ} is the probability of 0 successes, given that the mean number of successes is µ, etc. The probability of 1 or more successes is 1 - P(0; µ) = 1 - e^{-µ}. The distribution P(k; 1.67) is shown in Fig. A.6.
Figure A.6. The Poisson distribution P(k;1.67). The mean value is 1.67, the standard deviation 1.29. The curve is similar to the binomial distribution shown in Fig. A.4, but it is defined for values of k > 10. For example, P(20;1.67) = 2.2 x 10^{-15}. |