, ▁ ▁is ▁the ▁cum ulative ▁distribution ▁function ▁of ▁the ▁ ▁distribution ▁. ▁ ▁In ▁the ▁case ▁when ▁the ▁variance ▁ σ 2 ▁is ▁known , ▁ ▁is ▁optimal ▁in ▁terms ▁of ▁producing ▁the ▁short est ▁confidence ▁intervals ▁at ▁any ▁given ▁level . ▁In ▁the ▁case ▁when ▁the ▁variance ▁ σ 2 ▁is ▁unknown , ▁ ▁is ▁an ▁optimal ▁confidence ▁distribution ▁for ▁ μ . ▁ ▁Example ▁ 2 : ▁B ivari ate ▁normal ▁correlation ▁▁ ▁Let ▁ ρ ▁denotes ▁the ▁correlation ▁coefficient ▁of ▁a ▁b ivari ate ▁normal ▁population . ▁It ▁is ▁well ▁known ▁that ▁Fish er ' s ▁z ▁defined ▁by ▁the ▁Fish er ▁transformation : ▁ ▁has ▁the ▁limit ing ▁distribution ▁ ▁with ▁a ▁fast ▁rate ▁of ▁convergence , ▁where ▁r ▁is ▁the ▁sample ▁correlation ▁and ▁n ▁is ▁the ▁sample ▁size . ▁ ▁The ▁function ▁ ▁is ▁an ▁asympt otic ▁confidence ▁distribution ▁for ▁ ρ . ▁ ▁Using ▁CD ▁to ▁make ▁inference ▁ ▁Conf idence ▁interval ▁▁▁▁ ▁From ▁the ▁CD ▁definition , ▁it ▁is ▁evident ▁that ▁the ▁interval ▁ ▁and ▁ ▁provide ▁ 1 0 0 ( 1   −   α ) % - level ▁confidence ▁intervals ▁of ▁different ▁kinds , ▁for ▁ θ , ▁for ▁any ▁α   ∈   ( 0 ,   1 ). ▁ ▁Also ▁ ▁is ▁a ▁level ▁ 1 0 0 ( 1   −   α 1   −   α 2 ) % ▁confidence ▁interval ▁for ▁the ▁parameter ▁ θ ▁for ▁any ▁α 1   >   0 , ▁α 2