EXAMPLE 5.30

A gas station sells three grades of gasoline: regular, extra, and super. These are priced at \ {3.00},$ {3.20}$,and$$ {3.40}$pergallon,respectively.Let${X}{1},{X}{2}$,and${X}{3}$denotetheamountsofthesegradespurchased(gallons)onaparticularday.Supposethe${X}{i}$’sareindependentwith${\mu }{1} = {1000},{\mu }{2} = {500},{\mu }{3} = {300},{\sigma }{1} = {100},{\sigma }{2} = {80}$,and${\sigma }{3} = {50}$.Therevenuefromsalesis$Y = {3.0}{X}{1} + {3.2}{X}{2} + {3.4}{X}_{3}$ , and

$$E\left(Y\right)=3.0{\mu }_1+3.2{\mu }_2+3.4{\mu }_3=\$5620$$ $$V\left(Y\right)={\left(3.0\right)}^2{\sigma }_1^2+{\left(3.2\right)}^2{\sigma }_2^2+{\left(3.4\right)}^2{\sigma }_3^2=184,436$$ $${\sigma }_Y=\sqrt{184,436}=\$429.46$$