Performance evaluation of Monte Carlo simulation: Case study of Monte Carlo approximation vs. analytical solution for a chi-squared distribution

The guide to the expression of uncertainty in measurement (GUM) describes the law of propagation of uncertainty for linear models based on the first-order Taylor series approximation of Y = f(X1, X2, …, XN). However, for non-linear models this framework leads to unreliable results while estimating the combined standard uncertainty of the model output [u(y)]. In such instances, it is possible to implement the method(s) described in Supplement 1 to GUM – Propagation of distributions using a Monte Carlo Method. As such, a numerical solution is essential to overcome the complexity of the analytical approach to derive the probability density functions of the output. In this paper, Monte Carlo simulations are performed with the aim of providing an insight into the analytical transformation of the probability density function (PDF) for Y = X2 where X is normally distributed and a detailed comparison of analytical and Monte Carlo approach results are provided. This paper displays how the used approach enables to find PDF of Y = X2 without the use of special functions. In addition, the singularity of the PDF and the nonsymmetric coverage interval are also discussed