We　investigate　the　evaluation　of　structural　reliability　under　imperfect　knowledge　about　the　probability　distributions　of　random　variables,　with　emphasis　on　the　uncertainties　of　the　distribution　parameters.　When　these　uncertainties　are　considered,　the　failure　probability　becomes　a　random　variable　that　is　referred　to　as　the　conditional　failure　probability.　For　the　sake　of　transparency　in　communicating　risk,　it　is　necessary　to　determine　not　only　the　mean　but　also　the　quantile　of　the　conditional　failure　probability.　A　novel　method　is　proposed　for　estimating　the　quantile　of　the　conditional　failure　probability　by　using　the　probability　distribution　of　the　corresponding　conditional　reliability　index,　in　which　a　point-estimate　method　based　on　bivariate　dimension-reduction　integration　is　first　suggested　to　compute　the　first　three　moments　(i.e.,　mean,　standard　deviation　and　skewness)　of　the　conditional　reliability　index.　The　probability　distribution　of　the　conditional　reliability　index　is　then　approximated　by　a　three-parameter　square　normal　distribution.　Numerical　studies　show　that　the　computational　efficiency　of　the　proposed　method　was　well　above　that　of　Monte　Carlo　simulations　without　loss　of　accuracy,　and　also　show　that　neglecting　parameter　uncertainties　will　lead　to　the　structural　reliability　being　overestimated.　The　developed　methodology　provides　a　complete　picture　of　structural　reliability　evaluation　under　imperfect　knowledge　about　probability　distributions.　(C)　2018　Elsevier　Ltd.　All　rights　reserved.