#### Date of Award

Spring 1-1-2017

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Applied Mathematics

#### First Advisor

Gregory Beylkin

#### Second Advisor

Alireza Doostan

#### Third Advisor

Michael Sprague

#### Fourth Advisor

Bengt Fornberg

#### Fifth Advisor

Keith Julien

#### Abstract

In this thesis we construct novel functional representations for the Probability Density Functions (PDFs) of random variables and develop efficient and accurate algorithms for computing the PDFs of their sums, products and quotients, again in the same representation. We consider two important cases of random variables: non-negative random variables and random variables taking both positive and negative values. For the first case, we use approximations by decaying exponentials with complex exponents, while for the second case we develop a Gaussian-based multiresolution analysis (GMRA).

The need to represent distributions of products and quotients of random variables appear in many areas of theoretical and applied sciences. However, there are currently only limited number of numerical techniques for computing such products and quotients and this thesis presents new numerical methods for this purpose. Current methods for computing the product and quotients typically rely on a Monte Carlo type approach, where the PDFs of the product or quotient are sampled individually and the histogram of the resulting PDF is obtained from computed products or quotients of the individual samples. Although Monte Carlo methods are easy to implement, they suffer from slow convergence and therefore are not well suited for achieving high accuracy. Another method for computing the PDFs of the products and ratios of positive independent random variables relies on the Mellin transform and we describe such methods in greater detail in the thesis. Although mathematically appealing, techniques based on the Mellin transform lack in robust and stable numerical algorithms for computation of the inverse Mellin transform, hence making them not universally applicable.

Our novel representations and associated numerical algorithms produce a general framework for computing of PDFs of random variables which we call numerical calculus of PDFs in functional form. The new fast algorithms of this thesis allow user to select computational accuracy; the speed of algorithms only weakly depends on such selection. We demonstrate the performance of new algorithms on multiple examples using accuracies that are well beyond the reach of Monte Carlo based methods.

#### Recommended Citation

Satkauskas, Ignas V., "Numerical Calculus of Probability Density Functions" (2017). *Applied Mathematics Graduate Theses & Dissertations*. 93.

https://scholar.colorado.edu/appm_gradetds/93