i   ~   N ( μ ,   σ 2 ), ▁i   =   1 ,   2 ,   ...,   n ▁is ▁given . ▁ ▁( 1 ) ▁Var iance ▁ σ 2 ▁is ▁known ▁ ▁Let , ▁ Φ ▁ ▁be ▁the ▁cum ulative ▁distribution ▁function ▁of ▁the ▁standard ▁normal ▁distribution , ▁and ▁ ▁the ▁cum ulative ▁distribution ▁function ▁of ▁the ▁Student ▁ ▁distribution . ▁Both ▁the ▁functions ▁ ▁and ▁ ▁given ▁by ▁ ▁satisfy ▁the ▁two ▁requirements ▁in ▁the ▁CD ▁definition , ▁and ▁they ▁are ▁confidence ▁distribution ▁functions ▁for   μ . ▁ ▁Furthermore , ▁ ▁satisfies ▁the ▁definition ▁of ▁an ▁asympt otic ▁confidence ▁distribution ▁when ▁n → ∞ , ▁and ▁it ▁is ▁an ▁asympt otic ▁confidence ▁distribution ▁for ▁ μ . ▁The ▁uses ▁of ▁ ▁and ▁ ▁are ▁equivalent ▁to ▁state ▁that ▁we ▁use ▁ ▁and ▁ ▁to ▁estimate ▁, ▁respectively . ▁ ▁( 2 ) ▁Var iance ▁ σ 2 ▁is ▁unknown ▁ ▁For ▁the ▁parameter ▁ μ , ▁since ▁ ▁involves ▁the ▁unknown ▁parameter ▁ σ ▁and ▁it ▁viol ates ▁the ▁two ▁requirements ▁in ▁the ▁CD ▁definition , ▁it ▁is ▁no ▁longer ▁a ▁" distribution ▁estim ator " ▁or ▁a ▁confidence ▁distribution ▁for   μ . ▁However , ▁ ▁is ▁still ▁a ▁CD ▁for ▁ μ ▁and ▁ ▁is ▁an ▁a CD ▁for   μ . ▁ ▁For ▁the ▁parameter ▁ σ 2 , ▁the ▁sample - dependent ▁cum ulative ▁distribution ▁function ▁ ▁is ▁a ▁confidence ▁distribution ▁function ▁for ▁ σ 2 . ▁Here