ABSTRACT
The probability distribution function (pdf) of the Poisson (named after Simeon Poisson) distribution can be derived by taking the limit of the binomial pdf as n→ ∞, P → 0, and nP =μ remains constant. The Poisson pdf is given by
f x e
x x
( ) = = ¼ -mm
! , , , ,0 1 2 (8.1)
The term f(x) is the probability of x occurrences of an event that occurs on the average μ times per unit of space or time. Both the mean and the variance of a random variable X having a Poisson distribution are μ. Interestingly, the Poisson parameter is both the mean and variance of the distribution.1,2
Examining Equation 8.1, if any “incident” occurred at an average rate of λ incidents per time, then the probabilities of 0, 1, 2,… incidents occurring in any one unit of time would be expected to equal the successive terms in Equation 8.1 of the Poisson distribution, that is,
e-+ + + é
ë ê
ù
û ú
l l l l1 2 3
! ! (8.2)
Three bar charts of the Poisson distribution for various values of λ equal to 1, 2, and 5 are presented in Figure 8.1.
