 The shape parameter "α" is the tail index, "μ" is location, "σ" is scale, "γ" is an inequality parameter.Some special cases of Pareto Type (IV) are

The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index "α" (inequality index "γ").In particular, fractional "δ"-moments are finite for some "δ" > 0, as shown in the table below, where "δ" is not necessarily an integer.