PCM-2 Manual.pdf - Voss Associates

DISTRIBUTION FUNCTIONS
Binomial Distribution
The binomial distribution is quite cumbersome to use. For that reason, it is rarely invoked for
analysis of radiological counting exercises. It is worthy of mention because the popular Poisson and
Gaussian functions are derived from it.
Poisson Distribution
This distribution function is an approximation of the binomial distribution function and is applicable
when:
a.
A large sample size is involved, and
b.
The probability of success associated with a single element, or member of the
sample population, is very small.
The above criteria is applicable to most radiological counting exercises. The sample size, in terms of
number of atoms, usually will approach the order of magnitude of avogadro's number. The
probability that anyone atom will disintegrate within time t is expressed by Eq. 22.
Eq.22
Where A is the decay constant for the isotope of interest. Since the half life of many common
isotopes is very large when compared to typical count times, p is usually much less than 1.
The Poisson distribution function is best applied where fewer than 20 counts occur in a single count
cycle and is appropriate for many alpha radiation counters where background count rates of less than
10 counts per minute (cpm) are frequently encountered. Efficiencies (detector counts/disintegration)
of such instruments can be low since alpha particles are rapidly attenuated in air. Low efficiencies
coupled with low levels of activity that are of regulatory concern result in the "less-than-20-counts"
rule-of-thumb being satisfied for the "signal" part of the measurement or counting exercise.
It is noted that the classical definition of "measurement" makes the term inappropriate for radiological
instrumentation that count pulses. However, popular use of the term deem its incorporation in this
text as appropriate.
The basic function of the Poisson Distribution is expressed in Eq. 23.
e -~ AX
p(x) =
xl
Eq.23
Where:p(x) =
The probability that exactly x events will occur.
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