# 10_03 - STAT 410 Examples for mean variance 2 Fall 2011...

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STAT 410 Examples for 10/03/2011 Fall 2011 Population: mean μ , variance σ 2 , standard deviation σ . Random Sample: X 1 , X 2 , … , X n . or X 1 , X 2 , … , X n are i.i.d. E ( X 1 + X 2 + … + X n ) = n μ , Var ( X 1 + X 2 + … + X n ) = n σ 2 , SD ( X 1 + X 2 + … + X n ) = σ n . The sample mean n n X ... X X 2 1 X + + + = . E ( X ) = μ , Var ( X ) = n 2 σ , SD ( X ) = n σ . If the sampling is done without replacement from a finite population of size N , then SD ( Σ X ) = 1 N N n n σ - - , SD ( X ) = 1 N N n n σ - - .

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Let X 1 , X 2 , … , X n be i.i.d. 2 σ μ , N . Let n n n ... i X X X X X 2 1 + + + = = ( sample mean ) ( ) 1 X X S 2 2 - = - n i ( sample variance ) Then X and S 2 are independent; X has 2 σ μ , n N distribution; n σ μ X - has ( )

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10_03 - STAT 410 Examples for mean variance 2 Fall 2011...

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