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Unformatted text preview: Administration I have regraded midterms. Come get them after class. CS70: Satish Rao: Lecture 26. 1. Variance Calculations. 2. Independent random variables. 3. Variance Properties. Example Pr [ X = 1 ] = . 99 Pr [ X = 99 ] = . 01 Example Pr [ X = 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = 1 × . 99 + 99 × . 01 = . Example Pr [ X = 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = 1 × . 99 + 99 × . 01 = . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) ≈ 100 . Example Pr [ X = 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = 1 × . 99 + 99 × . 01 = . 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 = . E [ X 2 ] = 1 × . 99 +( 99 ) 2 × ( . 01 ) ≈ 100 . Var ( X 2 ) ≈ 100 = ⇒ σ ( X ) ≈ 10 . E (  X  ) = 1 × . 99 + 99 × . 01 = 1 . 98 . Example Pr [ X = 1 ] = . 99 Pr [ X = 99 ] = . 01 E [ X ] = 1 × . 99 + 99 × . 01 = . 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 = . 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. 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 + ... 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 + ... 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 + ......
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 Fall '11
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