▁the ▁variance ▁of ▁the ▁sample ▁paths , ▁impro ving ▁the ▁precision . ▁ ▁Suppose ▁that ▁we ▁would ▁like ▁to ▁estimate ▁▁ ▁For ▁that ▁we ▁have ▁generated ▁two ▁samples ▁ ▁An ▁un bi ased ▁estimate ▁of ▁ ▁is ▁given ▁by ▁ ▁And ▁▁ ▁so ▁variance ▁is ▁reduced ▁if ▁ ▁is ▁negative . ▁ ▁Example ▁ 1 ▁ ▁If ▁the ▁law ▁of ▁the ▁variable ▁X ▁follows ▁a ▁uniform ▁distribution ▁along ▁[ 0 ,   1 ], ▁the ▁first ▁sample ▁will ▁be ▁▁ ▁, ▁ ▁where , ▁for ▁any ▁given ▁i , ▁ ▁is ▁obtained ▁from ▁U ( 0 ,   1 ). ▁The ▁second ▁sample ▁is ▁built ▁from ▁▁ ▁, ▁ ▁where , ▁for ▁any ▁given ▁i : ▁. ▁▁ ▁If ▁the ▁set ▁ ▁is ▁uniform ▁along ▁[ 0 ,   1 ], ▁so ▁are ▁. ▁ ▁Furthermore , ▁cov ar iance ▁is ▁negative , ▁allowing ▁for ▁initial ▁variance ▁reduction . ▁ ▁Example ▁ 2 : ▁integral ▁calculation ▁ ▁We ▁would ▁like ▁to ▁estimate ▁ ▁The ▁exact ▁result ▁is ▁▁ ▁. ▁ ▁This ▁integral ▁can ▁be ▁seen ▁as ▁the ▁expected ▁value ▁of ▁ ▁, ▁ ▁where ▁ ▁and ▁U ▁follows ▁a ▁uniform ▁distribution   [ 0 ,   1 ]. ▁ ▁The ▁following ▁table ▁comp ares ▁the ▁classical ▁Monte ▁Carlo ▁estimate ▁( sample ▁size : ▁ 2 n , ▁where ▁n   =   1 5 0 0 ) ▁to ▁ ▁the ▁ant ith etic ▁vari ates ▁estimate ▁( sample ▁size : ▁n , ▁completed ▁with ▁the ▁transformed ▁sample ▁ 1   −   ui