13. Uncertainty Quantification

As a check, we can also derive the same result by directly manipulating Eq. (3.​1):

Given our assumptions about the model-input distributions, we know that \( \mathbb {V}[X_{1}]=\mathbb {V}[X_{2}]=\mathbb {V}[X_{3}]=\dfrac{1}{12}\) and \(\mathbb {E}[X_{1}]^{2}=\mathbb {E}[X_{1}]^{2}=\dfrac{1}{3}\). If we substitute into Eq. (13.1), we find

The prerequisite to obtaining x is the generation of uniformly distributed random numbers (the u’s). This is not an easy task, especially if we aim for very large sample sizes. There is a vast amount of literature on this subject, as a starting reference see Marsaglia and Zaman (1993). The main idea is to create sequences of numbers that depend on an initial seed, but then, within the sequence, carry the property of being uniformly distributed. Clearly, the sequence repeats if the seed is re-obtained. For example, a simple method is to start with a seed of four digits, \(s_{0}=5678\). Then, let \(a= \sqrt{3}*100000000\) and take the 12-digit numberAt this point, consider the four central digits, 5844 and repeat the process, setting \(s_{1}=5844\) and multiplying \(s_{1}\) times a.

The analyst should be aware that if she is asking for a very large sample size, she should not exceed the limit of the pseudo random number generator offered by the software in use. However, this is rarely the main preoccupation of an analyst performing a sensitivity analysis.