07_01 - Expectation p. 7-1 Recall. Expectation for...

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p. 7-1 Expectation • Recall . Expectation for univariate random variable. • Theorem. For random variables X =( X 1 , … , X n ) with joint pmf p X /pdf f X , the expectation of a univariate random variable Y , where Y = g ( X 1 , … , X n ), g : R n R 1 , is if X 1 , … , X n are discrete and the sum converges absolutely, or if Y and X 1 , … , X n are continuous and the integrals converges absolutely. Proof . Like the univariate case. Q : What if Y is discrete and X 1 , … , X n are continuous? E ( Y ) y Y yp Y ( y ) = x =( x 1 ,...,x n ) X g ( x 1 ,...,x n ) p X ( x 1 n ) E [ g ( X 1 ,...,X n )] (1) (2) (3) (4) E ( Y ) −∞ yf Y ( y ) dy = −∞ ··· −∞ g ( x 1 n ) f X ( x 1 n ) dx 1 dx n E [ g ( X 1 n )] p. 7-2 Notation. Shorthand notation. Combine (1) and (3), by writing Riemann-Stieltjes Integral. For example, for non-negative g , and combine (2) and (4) by writing where the limit is taken over all a = x 0 < x 1 < < x n = b as n 0 and [Recall . The integral of g over (a, b] is defined as max i =1 ,...,n ( x i x i 1 ) 0 . b a g ( x ) dF ( x )=lim n i =1 g ( x i )[ F ( x i ) F ( x i 1 )] . b a g ( x ) dx =lim n i =1 g ( x i )( x i x i 1 ) . ] Note. g ( X 1 , … , X n )= X i E [ g ( X 1 , … , X n )]= E ( X i ) g ( X 1 , … , X n )=( X i a aa ) 2 E [ g ( X 1 , … , X n )]= Var ( X i ) μ X i . μ X i σ 2 X i .
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p. 7-3 Example (Distance between two points). Suppose that X , Y are i.i.d. ~ Uniform(0, 1). Let D =| X Y |. Find E ( D ). f ( x,y )= 1 , 0 x 1, 0 y 1 , 0 , otherwise . The joint pdf of ( X , Y ) is • Theorem (Mean of Sum). For r.v.’s X 1 , … , X n and constants < a 0 , a 1 , …, a n < , E ( a 0 + a 1 X 1 + + a n X n ) = a 0 + a 1 E ( X 1 )+ + a n E ( X n ).
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07_01 - Expectation p. 7-1 Recall. Expectation for...

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