5.20. Treatment of numbers of counts and count rates

The feature for non-normal distributed numbers of counts and count rates to be described now refers to Monte Carlo simulations according to ISO 11929-2019, part 2. According to part 1 of ISO 11929:2019, the input quantities in any case are assumed as normal-distributed or are attributed to this distribution by the principle of maximum entropy.

According to the GUM Supplement 1 (JCGM 101:2008), clause 6.4.11, for counted events, which are Poisson distributed and represent an input quantity \(X\), e.g., counted photons, the following step is recommended to be taken for determining the distribution of the input quantity. If \(\mathbf{q}\) events are counted, a Gamma distribution is assigned to the posterior of the quantity \(\mathbf{X}\) by applying the Bayes theorem and using a constant prior. This is to be used as the distribution associated with \(X\):

(5.20.1)\[g_{X}(\xi) = \frac{\xi^{q}e^{- \xi}}{q!} \equiv Ga\left( \xi|q + 1,1 \right) \text{ for } \xi \geq 0\]

Mean and variance are \(E\lbrack X\rbrack = q + 1\) and \(Var\lbrack X\rbrack = q + 1\), respectively. This refers to numbers of counts.

For Poisson-distributed numbers of counts, a Gamma distribution is assigned In ISO 11929-2019 to the associated count rate \(\rho\), where a prior \(\rho^{- 1}\) is used instead of a constant prior. Mean and variance are in this case given as \(q/t\) und \(q/t^{2}\), respectively.

In UncertRadio, the described step is treated as follows. As already given by Eq. (5.20.1), the Gamma distribution is assigned to the number \(n\) of counts by selecting the distribution type „(N+x) rule“ for \(n\) (this corresponds to a prior \(\rho^{- 1}\)). By calculating the corresponding count rate \(R\), which requires an equation like \(R = n/t\), the count rate is also Gamma distributed. This also means, that a count rate to be treated in UncertRadio as Gamma distributed, always requires defining it by an equation like \(R = n/t\).

When measuring an activity, two variants are to be considered,

• a measurement with pre-selected counting duration (the registered number \(n\) is randomly distributed, following a Poisson distribution), and

• a measurement with pre-selected numbers of counts (the counting duration \(t\) is randomly distributed, following an Erlang distribution).

The Erlang distribution is addressed in the textbook by Knoll (Knoll, G.F., Radiation Detection and Measurement, 2nd edition, (John Wiley, NewYork,1989), pp. 96-99);

See also:

• International Safety Research, Safety Support Series, 2013. Radiation Counting Statistics. Volume 1. Canada.

• Pengra, D., 2008

• Pishro-Nik, H., Introduction to Probability

Comparing Erlang and Poisson distributions

The two distributions are defined as follows, with \(\rho\) designating the count rate parameter:

Poisson distribution:

(5.20.2)\[P_{Poi}(n) = \frac{(\rho t)^{n}e^{- \rho\ t}}{n!}\]

\[E\left\lbrack P_{poi} \right\rbrack = \ Var\left\lbrack P_{poi} \right\rbrack = \rho t\]

Erlang distribution:

(5.20.3)\[P_{Erl}(t) = \frac{\rho^{n}t^{n - 1}e^{- \rho\ t}}{(n - 1)!}\left\lbrack \equiv Ga\left( t|n,\rho \right) \right\rbrack\]

\[E\left\lbrack P_{Erl} \right\rbrack = n/\rho;\ \ \ \ \ \ \ \ \ Var\left\lbrack P_{poi} \right\rbrack = n/\rho^{2}\]

The Erlang distribution is a Gamma distribution for integer-valued \(n\). The two formulae ((5.20.2) and (5.20.3)) lead to a simple relation:

(5.20.4)\[{t\ P}_{Erl}\left( t|\rho,n \right) = n\ P_{poi}\left( n|\rho,t \right)\]

Applying the Bayes theorem with a prior \(\rho^{- 1}\) to both distributions results in the same posterior distribution for the count rate \(\rho\), a Gamma distribution:

Measurement with pre-set time:

(5.20.5)\[\frac{P_{Poi}\left( n|\rho,t \right)\rho^{- 1}}{\int_{}^{}{P_{Poi}\left( n|\rho,t \right)\rho^{- 1}d\rho}} = \frac{P_{Poi}\left( n|\rho,t \right)\rho^{- 1}}{1/n} = \frac{n(\rho t)^{n}e^{- \rho\ t}\rho^{- 1}}{n!} = \frac{t^{n}{\rho^{n - 1}e}^{- \rho\ t}}{(n - 1)!} = Ga\left( \rho|n,t \right)\]

Measurement with pre-set counts:

(5.20.6)\[\frac{P_{Erl}\left( t|\rho,n \right)\rho^{- 1}}{\int_{}^{}{P_{Erl}\left( t|\rho,n \right)\rho^{- 1}d\rho}} = \frac{P_{Erl}\left( t|\rho,n \right)\rho^{- 1}}{1/t} = \frac{t\rho^{n}t^{n - 1}e^{- \rho\ t}\rho^{- 1}}{(n - 1)!} = \frac{t^{n}{\rho^{n - 1}e}^{- \rho\ t}}{(n - 1)!} = Ga\left( \rho|n,t \right)\]

By equating the second parts of the two equations (5.20.5) and (5.20.6), the simple relation of (5.20.4) is obtained again:

\[\frac{P_{Poi}\left( n|\rho,t \right)\rho^{- 1}}{1/n} = \frac{P_{Erl}\left( t|\rho,n \right)\rho^{- 1}}{1/t}\]

or

(5.20.7)\[t\ P_{Erl}\left( t|\rho,n \right) = n\ P_{Poi}\left( n|\rho,t \right)\]

If another prior is used for the Poisson distribution, \(\rho^{- 1/2}\), again a Gamma distribution is obtained, but a different one: \(Ga\left( \rho|n + 1/2,t \right)\).

In the case of pre-set counts (math:t variable), the Erlang distribution must be assigned to \(t\) by selecting the distribution type “Npreset“ for \(t\). By an also required equation like \(R = n/t\), the Gamma distribution\(\ Ga\left( \rho|n,t \right)\) is thereby internally assigned to the count rate \(R\).

Example project: PresetCounts_EN.txp