On some properties of Cauchy-Stieltjes kernel families

We explore properties of Cauchy-Stieltjes Kernel (CSK) families that have the same counterpart in natural exponential families (NEFs). We determine the variance function of the finite mixtures of a CSK family with its length-biased family. We also prove that, for a more natural definition for the domain of means, the new domain of means scale nicely under affine transformation.