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CS70: Satish Rao: Lecture 26. 1. Variance Calculations. 2. Independent random variables. 3. Variance Properties.
Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01

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Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 .
Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) 100 .

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Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) 100 . Var ( X 2 ) 100 = σ ( X ) 10 .
Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) 100 . Var ( X 2 ) 100 = σ ( X ) 10 . E ( | X | ) = 1 × . 99 + 99 × . 01 = 1 . 98 .

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Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) 100 . Var ( X 2 ) 100 = σ ( X ) 10 . E ( | X | ) = 1 × . 99 + 99 × . 01 = 1 . 98 . Different by factor of 5.
Example Pr [ X = - 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = - 1 × . 99 + 99 × . 01 = 0 . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) 100 . Var ( X 2 ) 100 = σ ( X ) 10 . E ( | X | ) = 1 × . 99 + 99 × . 01 = 1 . 98 . Different by factor of 5. Exercise: make it bigger.

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Variance: geometric distribution. X is a geometrically distributed variable with parameter p .
Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ...

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Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ...
Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ...

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Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) - ( p + p ( 1 - p )+ p ( 1 - p ) 2 + ... )
Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) E ( X ) ! - ( p + p ( 1 - p )+ p ( 1 - p ) 2 + ... )

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Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) E ( X ) ! - ( p + p ( 1 - p )+ p ( 1 - p ) 2 + ... ) Distribution.
Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) E ( X ) ! - ( p + p ( 1 - p )+ p ( 1 - p ) 2 + ... ) Distribution. pE ( X 2 ) = 2 E ( X ) - 1

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Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) E ( X ) ! - ( p + p ( 1 - p )+ p ( 1 - p ) 2 + ... ) Distribution. pE ( X 2 ) = 2 E ( X ) - 1 = 2 ( 1 p ) - 1
Variance: geometric distribution. X is a geometrically distributed variable with parameter p . E ( X 2 ) = p + 4 p ( 1 - p )+ 9 p ( 1 - p ) 2 + ... - ( 1 - p ) E ( X 2 ) = p ( 1 - p )+ 4 p ( 1 - p ) 2 + ... pE ( X 2 ) = p + 3 p ( 1 - p )+ 5 p ( 1 - p ) 2 + ... = 2 ( p + 2 p ( 1 - p )+ 3 p ( 1 - p ) 2 + .. ) E ( X ) !

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