▁and ▁the ▁tail ▁behavior ▁are ▁of ▁particular ▁interest . ▁ ▁Other ▁families ▁of ▁distributions ▁can ▁be ▁used ▁if ▁the ▁focus ▁is ▁on ▁other ▁devi ations ▁from ▁normal ity . ▁ ▁If ▁the ▁symmetry ▁of ▁the ▁distribution ▁is ▁the ▁main ▁interest , ▁the ▁ske w ▁normal ▁family ▁or ▁version ▁ 2 ▁of ▁the ▁generalized ▁normal ▁family ▁discussed ▁below ▁can ▁be ▁used . ▁ ▁If ▁the ▁tail ▁behavior ▁is ▁the ▁main ▁interest , ▁the ▁student ▁t ▁family ▁can ▁be ▁used , ▁which ▁approxim ates ▁the ▁normal ▁distribution ▁as ▁the ▁degrees ▁of ▁freedom ▁grows ▁to ▁infinity . ▁ ▁The ▁t ▁distribution , ▁unlike ▁this ▁generalized ▁normal ▁distribution , ▁obtain s ▁heav ier ▁than ▁normal ▁t ails ▁without ▁acqu iring ▁a ▁c us p ▁at ▁the ▁origin . ▁ ▁Properties ▁ ▁M om ents ▁▁ ▁Let ▁ ▁be ▁zero ▁mean ▁generalized ▁Gaussian ▁distribution ▁of ▁shape ▁ ▁and ▁scaling ▁parameter ▁ ▁. ▁ ▁The ▁ ▁moments ▁of ▁▁ ▁exist ▁and ▁are ▁finite ▁for ▁any ▁k ▁ ▁greater ▁than ▁− 1 . ▁ ▁For ▁any ▁non - negative ▁integer ▁ ▁k , ▁the ▁plain ▁central ▁moments ▁are ▁ ▁Connection ▁to ▁Pos itive - Def in ite ▁Function s ▁▁ ▁The ▁probability ▁density ▁function ▁of ▁this ▁version ▁of ▁the ▁generalized ▁normal ▁distribution ▁is ▁a ▁positive - defin ite ▁function ▁ ▁for ▁ ▁. ▁ ▁In finite ▁divis ibility ▁▁ ▁This ▁version ▁of ▁generalized ▁Gaussian ▁distribution ▁is ▁an ▁infinitely ▁divis ible ▁distribution ▁if ▁and ▁only ▁if ▁. ▁ ▁General izations ▁ ▁The ▁mult ivari ate ▁generalized ▁normal ▁distribution , ▁i .