Distributions

Uniform distribution

The uniform distribution is defined on the interval [\(a\), \(b\)] where \(a\) and \(b\) are the minimum and maximum value and \(a>0\), \(b>0\).

The conversion factor is the nominal value of the uniform distribution, and

\[factor = \frac{a + b}{2}\]

For the variability upper value, \(b\) = \(upper\) and \(a\) is calculated as:

\[a = 2 \cdot factor - b\]

Lognormal distribution

The lognormal distribution is characterised by parameters \(\mu\) and \(\sigma\), which are the mean and standard deviation on log-scale.

The conversion factor is the nominal value of the lognormal distribution, \(\mu = ln(factor)\). The variability upper value specifies the p95 of the standard lognormal. Parameter \(\sigma\) is calculated as:

\[\sigma = \frac{ln(upper) - \mu}{1.645}\]

Beta distribution

The standard beta distribution is defined on the interval (0, 1) and is usually characterised by two parameters \(a\) and \(b\), with \(a>0\), \(b>0\) (see e.g. Mood et al. (1974)). Alternatively, it can be parameterised by the mean (conversion factor)

\[\mu = a/(a+b) \in (0, 1)\]

and the variance

\[\sigma^2 = ab/(a+b+1)^{-1}(a+b)^{-2}\]

Note that:

\[\frac{ab}{(a+b)^2 (a + b + 1)} = \frac{\mu(1-\mu)}{(a+b)^2 (a + b + 1)} < \frac{\mu(1-\mu)}{1} = \mu(1-\mu) \in (0. 0.5^2)\]

The conversion factor is the nominal value of the beta distribution, here the mean. The variability upper value is the variance of the beta distribution. The equations solved for \(a\) and \(b\) show that:

\[a = \left( \frac{1 - \mu}{\sigma^2} - \frac{1}{\mu}\right) \cdot \mu^2\]

and

\[b = a \cdot \left(\frac{1}{\mu} - 1 \right)\]

InverseUniform distribution

The inverse uniform distribution is defined on the interval [\(a\), \(b\)] where \(a\) and \(b\) are the minimum and maximum value and \(a>0\), \(b>0\).

The conversion factor is the median (nominal) value of the inverse uniform distribution, and the median is equal to

\[factor = \frac{2}{a+b}\]

The variability upper value is equal to \(\frac{1}{a}\), so

\[a = \frac{1}{upper}\]

and

\[b = \frac{2}{factor} - a\]

Uniform distributed values are drawn from the interval [\(a\), \(b\)] and inverted. See e.g. Invers distribution and Inverse transform.