Link and others have derived formulas for the moments of certain discrete probability distributions using various finite-difference operators. In this article, we establish a method of obtaining moments of certain discrete distributions that makes use of differential operators. The derivations are elegant and more compact than derivations using standard methods. Students in a first-year mathematical statistics course are typically more acquainted with differential operators than with difference operators. Contrasted with difference operators, this familiarity affords a better opportunity to introduce the connection between operator mathematics and probability in such courses. However, we propose that both operators be employed: differential operators to introduce the connection and difference operators to broaden the students' exposure and prepare them for such courses as time series analysis.