Riskiness in gambles that belong to the same location-scale family and with well-defined means and variances

This paper investigates the orderings of gambles with well-defined means and variances that belong to the same location-scale family under: (i) the index of riskiness analyzed by Aumann and Serrano (AS) and (ii) the measure of riskiness proposed by Foster and Hart (FH). For both measures we study the characteristics and properties of the representation on the (mean, variance) plane of the curves that include gambles with the same level of riskiness. Our analysis applies to gambles with truncated normal distributions and beta distributions. We also discuss the relationships between different riskiness measures derived from the AS and FH measures.