Chap4 MTH3401 part3 - Chapter 4 Mathematical Expectation...

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Chap 4 Mathematical Expectation 1 Chapter 4 Mathematical Expectation
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Chap 4 Mathematical Expectation 2 Means and Variance of Linear Combinations of Random Variables Theorem 4.10: If X and Y are random variables with joint probability distribution f ( x , y ), then Proof . 2 2 2 2 2 2 XY Y X bY aX ab b a σ σ σ σ + + = + XY Y X Y X Y X Y X Y X bY aX Y X bY aX bY aX bY aX ab b a Y X abE Y E b X E a Y b X a E b a bY aX E b a Y bE X aE bY aX E bY aX E σ σ σ μ μ μ μ μ μ μ μ σ μ μ μ μ σ 2 )] )( [( 2 ] ) [( ] ) [( } )] ( ) ( {[ } )] ( ) {[( ) ( ) ( ) ( } ] ) {[( 2 2 2 2 2 2 2 2 2 2 2 2 2 + + = - - + - + - = - + - = + - + = + = + = + = - + = + + + +
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Chap 4 Mathematical Expectation 3 Means and Variance of Linear Combinations of Random Variables Corollary 1: If X and Y are independent random variables, then Corollary 2: If X and Y are independent random variables, then Corollary 3: If X 1 , X 2 , …, X n are independent random variables, then . 2 2 2 2 2 Y X bY aX b a σ σ σ + = + . 2 2 2 2 2 2 2 1 2 2 2 2 1 1 1 n X n X X n X n a X a X a a a a σ σ σ σ + + + = + + + . 0
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