Variance and Standard Deviation - Penn Math

1 Variance and Standard DeviationChristopher CrokeUniversity of PennsylvaniaMath 115 UPenn, Fall 2011 Christopher CrokeCalculus 115 VarianceThe first first important number describing a probabilitydistribution is the mean or expected valueE(X).The next one is thevarianceVar(X) = 2(X). The square root ofthe Variance is called theStandard (xi) is the probability distribution function for a randomvariable with range{x1,x2,x3,..}and mean =E(X) then:Var(X) = 2= (x1 )2f(x1) + (x2 )2f(x2) + (x3 )2f(x3) +..It is a description of how the distribution spreads .NoteVar(X) =E((X )2).The Standard Deviation has the same units asX.

2 ( ifXismeasured in feet then so is .)Christopher CrokeCalculus 115 VarianceThe first first important number describing a probabilitydistribution is the mean or expected valueE(X).The next one is thevarianceVar(X) = 2(X). The square root ofthe Variance is called theStandard (xi) is the probability distribution function for a randomvariable with range{x1,x2,x3,..}and mean =E(X) then:Var(X) = 2= (x1 )2f(x1) + (x2 )2f(x2) + (x3 )2f(x3) +..It is a description of how the distribution spreads .NoteVar(X) =E((X )2).The Standard Deviation has the same units asX. ( ifXismeasured in feet then so is.)

3 Christopher CrokeCalculus 115 VarianceThe first first important number describing a probabilitydistribution is the mean or expected valueE(X).The next one is thevarianceVar(X) = 2(X). The square root ofthe Variance is called theStandard (xi) is the probability distribution function for a randomvariable with range{x1,x2,x3,..}and mean =E(X) then:Var(X) = 2= (x1 )2f(x1) + (x2 )2f(x2) + (x3 )2f(x3) +..It is a description of how the distribution spreads .NoteVar(X) =E((X )2).The Standard Deviation has the same units asX. ( ifXismeasured in feet then so is .)Christopher CrokeCalculus 115 VarianceThe first first important number describing a probabilitydistribution is the mean or expected valueE(X).

4 The next one is thevarianceVar(X) = 2(X). The square root ofthe Variance is called theStandard (xi) is the probability distribution function for a randomvariable with range{x1,x2,x3,..}and mean =E(X) then:Var(X) = 2= (x1 )2f(x1) + (x2 )2f(x2) + (x3 )2f(x3) +..It is a description of how the distribution spreads .NoteVar(X) =E((X )2).The Standard Deviation has the same units asX. ( ifXismeasured in feet then so is .)Christopher CrokeCalculus 115 VarianceThe first first important number describing a probabilitydistribution is the mean or expected valueE(X).The next one is thevarianceVar(X) = 2(X).

5 The square root ofthe Variance is called theStandard (xi) is the probability distribution function for a randomvariable with range{x1,x2,x3,..}and mean =E(X) then:Var(X) = 2= (x1 )2f(x1) + (x2 )2f(x2) + (x3 )2f(x3) +..It is a description of how the distribution spreads .NoteVar(X) =E((X )2).The Standard Deviation has the same units asX. ( ifXismeasured in feet then so is .)Christopher CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?

6 Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls.

7 The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?

8 Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls.

9 The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?

10 Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls. The first player is paid $2 if hewins but the second player gets $3 if she wins. No one gets payedif 4 white balls are have seen that the payout and probabilities for the first playerare:Payout Probability28150115 3615 The expected value was = is the Variance ?Alternative formula for Variance : 2=E(X2) ((X )2) =E(X2 2 X+ 2)=E(X2) +E( 2 X) +E( 2)=E(X2) 2 E(X) + 2=E(X2) 2 2+ 2=E(X2) CrokeCalculus 115 Problem:Remember the game where players pick balls from anurn with 4 white and 2 red balls.