This communication provides a derivation of the often-used oneparameter exponential family of distributions based on an infinite sine series. The main results associated with this derivation are the finite approximations of the probability density function (pdf) and the cumulative distribution function (cdf) of the exact one-parameter exponential family of distributions. The limit of these finite functions are the exact pdf and cdf. Numerical examples are provided to compare and contrast the finite approximations of distributions in terms of error using the finite-based cdf and the exact standard exponential cdf. We would also note that the finite approximations of the pdf and cdf vary in terms of both shape and percentage points. In view of this, the finite pdf and cdf offer a user the flexibility to potentially provide more accurate approximations in the context of fitting distributions to data rather than approximations that are based solely on the exact exponential pdf or cdf – most notably when distributions are heavy-(right)tailed.
Headrick, Todd C. and May, Michael E. "Derivation of the Exponential Distribution through an Infinite Sine Series." Applied Mathematical Sciences 11 (Jan 2017): 2023-2030. doi:10.12988/ams.2017.76197.