3. Univariate Continuous Distributions

Attribution

This chapter was written by Patricia Mares Nasarre, Robert Lanzafame, and Max Ramgraber. Find out more here.

Civil engineering and the Geosciences involve working with nature which comes with unavoidable uncertainties. Natural phenomena have an intrinsic aleatoric uncertainty that we need to model, however, we don’t always have the resources to take into account all possible scenarios. For example, it is impractical to design a system to protect against the largest imaginable flood, earthquake, chemical spill, climate change or sea level rise scenario. As a consequence, we often need to make decisions for uncertain conditions and data-scarce situations.

To this end, statistics and its key ingredient probability are indispensable. Continuous probability distributions, also known as probability density functions (PDFs) model this uncertainty explicitly, allowing us to quantify the probability associated with specific outcomes of our decisions, and to incorporate it rigorously in our decision-making.

To accomplish this, we first return to the fundamental concepts for describing probability for univariate continuous random variables and their distributions: the PDF and CDF. After using the Gaussian distribution to illustrate, the empirical distribution will be introduced. Observing that the Gaussian distribution cannot be used to model all uncertain random variables encountered science and engineering, we consider a wide range of non-Gaussian distributions, in particular those that can be described by relatively simple equations (i.e., parametric distributions), as well as a special form of parameterization used in Python (location, shape and scale used by scipy.stats Python module). We close by considering qualitative an quantitative approaches for modelling data with parametric distributions, including tools for verification and validation of the models: Goodness of Fit (GOF).