PCM-2 Manual.pdf - Voss Associates

x = An integer.
l\=
The mean number of events that occurred or are expected to occur. Popular
convention dictates use of the symbol A which is not to be confused with its use to
symbolize decay constant as in Eq. 22.
xl =x (x - 1) (x - 2) (x - 3) ... 1.
O! is defines as equal to 1.
Of greater interest than probabilities associated with discreet integers is a cumulative distribution, a
probability associated with "x or more events." In this text, upper case "P" is used to indicate a
cumulative probability. Eq. 24 illustrates the cumulative distribution P(x).
P(x) = LPx
;=x
Eq. 24
The Poisson Distribution Function is normalized, i.e., the sum of all probabilities is exactly 1.
Therefore, anyone who would attempt the summation of Eq. 24 would find relief in the identity of
Eq.25.
P(x) = 1 - [p(O) + p(l) + p(2) + ... p(x - 1)]
Eq. 25
EXAMPLE: A counter observes an average of 2.5 counts per interval. The probability distribution
for this mean value follows.
e -2.52.5 0
p(O) .082085 Eq. 26
O!
e -2.52.5 1
p(l) = 0.205212 Eq. 27
11
A convenient identity is:
e -2.52.52
p(2) = 0.256516 Eq. 28
21
p(n) = p(n - 1) ~ Eq. 29
n
The discrete and cumulative probability distribution is summarized in Table 1 for values through x =
10. Figure 2 is a histogram that graphically illustrates the distribution described by A=2.5. Note
that the histogram is skewed to the left. This is because the function's domain is non-negative
PCM2.MAf\