Solutions_to_Sample_Final_Exam

Solutions_to_Sample_Final_Exam - EL630 Solutions to Sample...

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Unformatted text preview: EL630 Solutions to Sample Final Exam 1. Box 1 contains 3 white, 2 red and 5 black balls. Box 2 contains 4 white, 4 red and 2 black balls. At random, we pick up one ball from each box. Find the probability that these two balls are in different color. : (They are in the same color) P Solution (Both are white)+ (Both are red)+ (Both are black) (3/10)(4/10) (2/10)(4/10) (5/10)(2/10) 30/100 0.3 (Two are in different color) 1- 0.3=0.7 P P P P = = + + = = = 2. X is uniformly distributed on [ /2, / 2], tan Y X - = . Find ( ) f y and draw it. Solution : 1 ' 2 2 ( ) tan( ), tan , ( ) 1/cos 1 y g x x x y g x x y- = = = = = + . ' 2 ( ) 1 ( ) | ( )| (1 ) X Y f x f y g x y = = + 3. X and Y are independent with exponential densities ( ) ( ), ( ) ( ), 3 . y x X Y f x e u x f y e u y -- = = Find the density and distribution of Z = X+3Y . Solution : y ( ) Y f y 1/ 1 ( / 3) ( /3) ( ) ( / 3) ( / 3) ( / 3) 3 , , 3 , ( ) (1/3) ( /3) ( /3) ( /3) ( /3) ( ) ( ) ( ) ( ) ( )( /3) ( ) ( /3) ( ) ( /3) (1/( /3)) | w w W Y Z X W z w w z z w z w z Z X Y Z X W W Y f w f w e u w e u w f z f z w f w dw e u z w e u w dw e e dwu z e e -- - -------- = + = + = = = = =- =- = =- ( / 3) ( ) ( ) ( ) 3 z z u z e e u z -- =-- ( / 3) ( /3) ( /3) ( ) ( /(3 ))(( 3/ ) | (1/ ) | ) ( ) ( /(3 ))(3/ (3/ ) (1/ ) 1/ ) ( ) /3 {1 ( )} ( ) /3 /3 x z x z Z z z z z F z e e u z e e u z e e u z ------ =-- + =-- +- =---- 4. X and Y are independent with exponential densities ( ) ( ), ( ) ( ). y x X Y f x e u x f y e u y-- = = Find the joint density Z = X+Y and W=X/Y . Solution : , / , z x y w x y = + = 0, 0, x y z w . /( 1) is the unique solution. /( 1) x wz w y z w = + = + z ( ) Z f z 2 2 2 2 1 1 | ( , )| | | | ( / 1/ )| ( 1) / 1/ / J x y x y y w z y x y = = - + = +- } , indep. ( ) 2 2 ( , ) ( , )/| | ( 1) ( 1) X Y x y z ZW XY z z f z w f x y J e e w w- +- = = = + + 2 1 ( , ) ( ) ( ) ( 1) z ZW f z w ze u z u w w- = + We see that Z, W are independent. (It is easy to check that 2 1 ( ) and ( ) ( 1) z ze u z u w w- + are density functions.) 5....
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Solutions_to_Sample_Final_Exam - EL630 Solutions to Sample...

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