On Surprise Indices Related to Univariate Discrete and Continuous Distributions: A Survey

The notion that the occurrence of an event is surprising has been discussed in the literature without adequate details. By definition, a surprise index is an index by which how surprising an event is may be determined. Since its inception, this index has been evaluated for univariate discrete probability models, such as the binomial, negative binomial, and Poisson probability distributions. In this article, we derive and discuss using numerical studies, in addition to the above-mentioned probability models, surprise indices for several other univariate discrete probability models, such as the zero-truncated Poisson, geometric, Hermite, and Skellam distributions, by adopting a established strategy and using the Mathematica, version 12 software. In addition, we provide symbolical expressions for the surprise index for several univariate continuous probability models, which has not been previously discussed. For illustrative purposes, we present some possible real-life applications of this index and potential challenges to extending the notion of the surprise index to bivariate and higher dimensions, which might involve ubiquitous normalizing constants.