hw4sol - Engineering 0020 Probability and Statistics for...

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Engineering 0020: Probability and Statistics for Engineers 1 Spring, 2007; Solutions: Homework #4 8. a y f(y) 5 10 b 1 )} 5 . 0 2 ( ) 2 4 {( } 0 ) 50 / 25 {( | )} 50 / ( ) 5 / 2 {( | ) 50 / ( } ) 25 / 1 ( ) 5 / 2 {( ) 25 / 1 ( ) ( 10 5 2 5 0 2 10 5 5 0 = + = + = + = y y y dy y ydy dy y f c P(Y 3) = 18 . 0 50 / 9 | } 50 / { ) 25 / 1 ( ) ( 3 0 2 3 0 3 = = = = y ydy dy y f d 92 . 0 )} 5 . 0 2 ( ) 48 . 1 2 . 3 {( } 0 5 . 0 { | )} 50 / ( ) 5 / 2 {( | ) 50 / ( } ) 25 / 1 ( ) 5 / 2 {( ) 25 / 1 ( ) ( ) 8 ( P 8 5 2 5 0 2 8 5 5 0 8 = + = + = + = = y y y dy y ydy dy y f Y e P(3 Y 8) = P(Y 8) - P(Y 3) = .74 f P(Y<2 Y>6) = 4 . 0 } ) 25 / 1 ( ) 5 / 2 {( ) 25 / 1 ( ) ( ) ( 10 6 2 0 6 2 = + = + dy y ydy dy y f dy y f 18. a x = (100 p ) th percentile implies that P(X x ) = ( x -A)/(B-A) = p , i.e., x = A + (B-A) p b E(X) = B A B A A B x dx A B x dx x xf | )} ( 2 / { )} /( 1 { ) ( 2 = = = ( B 2 -A 2 )/2( B-A ) = ( B+A )( B-A )/2( B-A ) = ( B+A )/2 E(X 2 ) = B A B A A B x dx A B x dx x f x | )} ( 3 / { )} /( 1 { ) ( 3 2 2 = = = ( B 3 -A 3 ) ÷ 3( B-A ) = ( B 2 +A 2 +AB )( B-A ) ÷ 3( B-A ) = ( B 2 +A 2 +AB )/3 V(X) = E(X 2 ) – {E(X)] 2 = ( A 2 +AB+B 2 )/3 - {( A+B )/2} 2 = ( B-A ) 2 /12 σ x = (B-A)/(12) 0.5 c E(X n ) = B A n B A n n A B n x dx A B x dx x f x | )} )( 1 /( { )} /( 1 { ) ( 1 + = = +
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hw4sol - Engineering 0020 Probability and Statistics for...

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