chapter6 - Functions of Random Variables Method of...

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Functions of Random Variables
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Method of Distribution Functions X 1 ,…,X n ~ f(x 1 ,…,x n ) • U=g(X 1 ,…,X n ) – Want to obtain f U (u) • Find values in (x 1 ,…,x n ) space where U=u Find region where U≤u Obtain F U (u)=P(U≤u) by integrating f(x 1 , …,x n ) over the region where U≤u • f U (u) = dF U (u)/du
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Example – Uniform X Stores located on a linear city with density f(x)=0.05 -10 ≤ x ≤ 10, 0 otherwise Courier incurs a cost of U=16X 2 when she delivers to a store located at X (her office is located at 0) 1600 0 80 ) ( ) ( 1600 0 40 4 4 05 . 0 05 . 0 ) ( ) ( 4 4 4 16 2 / 1 4 4 2 = = = - - = = = - ± = = = - - u u du u dF u f u u u u dx u U P u F u X u u U u X u X u U U U u u U
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Example – Sum of Exponentials X 1 , X 2 independent Exponential( θ ) f(x i )= θ -1 e -xi/ θ x i >0, θ >0, i=1,2 f(x 1 ,x 2 )= θ -2 e -(x1+x2)/ θ x 1 ,x 2 >0 U=X 1 +X 2 ( 29 [ ] ( 29 ) , 2 ( ~ 0 1 1 1 ) ( 1 1 1 1 1 1 1 1 ) ( , / 2 / 2 / / / / 2 / ) ( 0 2 / 0 2 / ) ( / 0 2 0 / / 0 2 1 / / 0 0 2 2 1 2 2 1 2 1 2 1 2 2 2 2 2 2 1 2 2 1 2 θ β α = = = - - = - - = - = - = - = = - + - = = + = - - - - - - + - - - - - - - - - - - - ∫ ∫ Gamma U u ue e u e e u f ue e dx e dx e dx e e dx e e dx dx e e u U P X u X u X u X X u U x u X u X X u U u u u u U u u x u x u x u x u x u x u x x u x x u x u
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Method of Transformations X~f X (x) U=h(X) is either increasing or decreasing in X f U (u) = f X (x)|dx/du| where x=h -1 (u) Can be extended to functions of more than one random variable: U 1 =h 1 (X 1 ,X 2 ), U 2 =h 2 (X 1 ,X 2 ), X 1 =h 1 -1 (U 1 ,U 2 ), X 2 =h 2 -1 (U 1 ,U 2 ) - = = - = = 2 2 1 1 2 1 2 1 1 2 2 1 2 2 1 1 2 2 1 2 2 1 1 1 ) , ( ) ( | | ) , ( ) , ( | | 1 du u u f u f J x x f u u f dU dX dU dX dU dX dU dX dU dX dU dX dU dX dU dX J U
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Example • f X (x) = 2x 0≤ x ≤ 1, 0 otherwise U=10+500X (increasing in x) x=(u-10)/500 f X (x) = 2x = 2(u-10)/500 = (u-10)/250 dx/du = d((u-10)/500)/du = 1/500 • f U (u) = [(u-10)/250]|1/500| = (u-10)/125000 10 ≤ u ≤ 510, 0 otherwise
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Method of Conditioning • U=h(X 1 ,X 2 ) • Find f(u|x 2 ) by transformations (Fixing X 2 =x 2 ) • Obtain the joint density of U, X 2 : • f(u,x 2 ) = f(u|x 2 )f(x 2 ) Obtain the marginal distribution of U by integrating joint density over X 2 - = 2 2 2 ) ( ) | ( ) ( dx x f x u f u f U
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Example (Problem 6.11) X 1 ~Beta( α= 2,β= 229 X 2 ~Beta( α= 3,β= 129 Independent U=X 1 X 2 Fix X 2 =x 2 and get f(u|x 2 ) ( 29 ( 29 1 0 )) ln( 1 ( 18 ) ln( 18 18 0 18 ) ln( 18 18 18 18 ) ( ) | ( ) ( 1 0 1 18 3 1 ) / 1 )( / ( 6 ) ( ) | ( ) , ( 0 1 ) / 1 )( / ( 6 ) | ( / 1 / 1 0 3 ) ( 1 0 ) 1 ( 6 ) ( 2 2 1 2 2 2 2 1 2 2 1 2 2 2 2 2 2
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chapter6 - Functions of Random Variables Method of...

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