Cumulative probability for the sum of exponentially-distributed variables

Exponential distributions appear in a wide range of applications including chemistry, nuclear physics, time series analyses, and stock market trends. There are conceivable circumstances in which one would be interested in the cumulative probability distribution of the sum of some number of exponential variables, with potentially differing constants in their exponents. In this article we present a pedagogical derivation of the cumulative distribution, which reproduces the known formula from power density analyses in the limit that all of the constants are equal, and which assumes no prior knowledge of combinatorics except for some of the properties of a class of symmetric polynomials in $n$ variables (Schur polynomials).