This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Spring '12
 ProfCampssi

Click to edit the document details