Appendix A Algorithms
270
A. Algorithms
Probability Theory
Counting Statistics
Confidence Levels
Radioactive decay is a randomly occurring event and must be described in statistical
terms. Any count of radiation emitted is subject to statistical fluctuation due to the
random nature of radioactive decay and due to the half-life of the radionuclide.
The sampling distribution of a series of random radioactive events can be described
as a Binomial distribution where the outcome of an event can be described as success
or failure.
The Poisson distribution is a special case of Binomial distribution and predicts well
outcome of counting measurements when the expected number of events is small.
The Poisson distribution is skewed for small number of events and getting more
symmetric as number of events increases.
The Binomial and Poisson distributions tend to approach the shape of normal
distribution when the probability of decay gets small and number of events gets >30.
Unlike the Binomial and Poisson distributions which are discrete, the Gaussian is
continuous probability distribution by a probability density function below:
Equation 7 Gaussian Probability Density Function
2
2
2
)x(
e
2
1
)x(G
σ
µ
πσ
−
−
=
(7)
… where
and x is an observation from a normal distribution
that has a mean μ and a standard deviation σ.
The normal or Gaussian distribution is symmetric, bell shaped and centered on μ. The
spread is described by standard deviation σ and the variance is characterized by σ
2
.
The mean, median and mode for the normal distribution are the same and are given
by:
Equation 8 Mean of Normal Distribution
∫
+∞
∞−
== 1dx)x(xG
µ
(8)
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