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As noted above, the Poisson distribution is obtained as an asymptotic limit of the binomial distribution when p is very small. The result is

Equation A.32 (A.32)

where µ = np, as before, and 0! is understood to be 1. This distribution is determined by one rather than two constants: sigma = (npq)1/2, but q = 1 - p appeq 1, so sigma = (np)1/2 = µ1/2. The standard deviation is equal to the square-root of the mean. The Poisson distribution is discrete: P(0; µ) = e 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. The distribution P(k; 1.67) is shown in Fig. A.6.

Figure 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.

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