  This paper presents a new framework for analysing forensic DNA samples using
probabilistic genotyping. Specifically it presents a mathematical framework for
specifying and combining the steps in producing forensic casework
electropherograms of short tandem repeat loci from DNA samples. It is
applicable to both high and low template DNA samples, that is, samples
containing either high or low amounts DNA. A specific model is developed within
the framework, by way of particular modelling assumptions and approximations,
and its interpretive power presented on examples using simulated data and data
from a publicly available dataset. The framework relies heavily on the use of
univariate and multivariate probability generating functions. It is shown that
these provide a succinct and elegant mathematical scaffolding to model the key
steps in the process. A significant development in this paper is that of new
numerical methods for accurately and efficiently evaluating the probability
distribution of amplicons arising from the polymerase chain reaction process,
which is modelled as a discrete multi-type branching process. Source code in
the scripting languages Python, R and Julia is provided for illustration of
these methods. These new developments will be of general interest to persons
working outside the province of forensic DNA interpretation that this paper
focuses on.
