Univariate continuous distributions: symmetries and transformations

Jones, M. C. (2015). Univariate continuous distributions: symmetries and transformations. Journal of Statistical Planning and Inference, 161 pp. 119–124.

DOI: https://doi.org/10.1016/j.jspi.2014.12.011


If the univariate random variable X follows the distribution with distribution function F, then so does Y=F−1(1−F(X)). This known result defines the type of (generalised) symmetry of F, which is here referred to as T-symmetry; for example, ordinary symmetry about θ corresponds to Y=2θ−X. Some distributions, with density fS, display a density-level symmetry of the form fS(x)=fS(s(x)), for some decreasing transformation function s(x); I call this S-symmetry. The main aim of this article is to introduce the S-symmetric dual of any (necessarily T-symmetric) F, and to explore the consequences thereof. Chief amongst these are the connections between the random variables following F and fS, and relationships between measures of ordinary symmetry based on quantiles and on density values.

Viewing alternatives


Public Attention

Altmetrics from Altmetric

Number of Citations

Citations from Dimensions
No digital document available to download for this item

Item Actions