Ch3 - Ch.3 Homework Solution 3-5 (Buffons Neddle Problem) A...

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Unformatted text preview: Ch.3 Homework Solution 3-5 (Buffons Neddle Problem) A needle of length L is dropped randomly on a plane ruled with parallel lines that are a distance D apart , where D L. Show that the probability that the needle comes to rest crossing a line is 2L/ D.Explain how this gives a mechanical means of estimating the value of . Sol: Y : Y ~Uniform(0,D) : X H * ~Uniform(0,2 ) -- = = 2 sin 2 sin 2 2 1 2 1 ) ( l D l D l dyd D x * P P( & * )= D l 2 3-6 A point is chosen randomly in the interior of an ellipse: 1 2 2 2 2 = + b y a x .Find the marginal densities of the x and y coordinates of the point . Sol: -- =- =- a a a a ab dx x a a b dx x a a b 2 2 2 2 2 2 ab y x f XY 1 ) , ( = , 1 2 2 2 2 < + b y a x ---- = = + +- =- 2 2 2 2 1 ) ( ), , sin 2 2 ( 1 2 2 2 2 2 u a a b u a a b X dy ab x f a c a u a u a u dx u a 3-8 Let X and Y have the joint density 1 , 1 , ) ( 7 6 ) , ( 2 + = y x y x y x f (a) By integrating over the appropriate regions , find (i)P(X>Y),(ii)P(X+Y 1),(iii)P(X ). Sol: 1 , 1 , ) ( 7 6 ) , ( 2 + = y x y x y x f (a)(i) 2 1 ) ( 7 2 ) ( 7 6 ) ( 1 3 2 1 = = = + = + = dx y x y y x dydx y x Y X P x (ii) - + = + x dydx y x Y X P 1 2 1 ) ( 7 6 ) 1 ( (iii) dxdy y x X P 2 2 1 1 ) ( 7 6 ) 2 1 ( + = 3-9 Suppose that (X,Y) is uniformly distributed over the region defined by 0 y 2 1 x- and -1 x 1 (a) Find the marginal densities of X and Y . (b) Find the two conditional density Sol: y 2 1 x- and -1 x 1 , 3 4 ) 1 ( 1 1 2 = +- - dx x , 4 3 ) , ( = y x f XY (a) 1 , 1 2 3 1 4 3 1 4 3 4 3 ) ( 1 1 , 4 3 4 3 ) 1 ( 4 3 4 3 ) ( 1 1 2 2 1 2 - =- +- = = - +- = +- = = --- +- y y y y dx y f x x x dy x f y y Y x X (b) 1 , 1 1 4 3 4 3 4 3 ) ( 1 1 , 1 2 1 2 1 3 4 3 ) ( 2 2 +- = +- = -- =- = y x x x y f x y y y x f X Y Y X 3-11 Let 1 U , 2 U and 3 U be independent random variables uniform on [0,1].Find the probability that the roots of the quadratic 3 2 2 1 U x U x U + + are real . Sol: 2 ln 6 1 36 5 ) 4 ( 3 1 1 4 1 4 1 1 2 2 3 1 4 1 1 1 2 2 3 1 2 2 3 3 1 3 1 + = + = du du du du du du U U U P u u u u u 2 ln 6 1 36 5 2 1 1 1 4 4 3 2 1 1 4 1 3 2 2 1 2 2 2 2 + = + du du du du du du u u u u 3-120 Question0 Let x y x x e y x c y x f x < - < - =- , , ) ( ) , ( 2 2 (a)Find c . (b)Find the marginal densities. 0 Solution0 (a) 8 1 1 3 ! 3 4 ) 4 ( 3 4 3 4 ) 3 1 ( ) ( ) ( , , ) ( ) , ( 3 3 2 2 2 2 2 2 2 = = - = = =- = =- =- =- - - = - - -- --- c c c dx x e c dx x y x y y y x ce dydx y x ce dydx e y x c x y x x e y x c y x f x x x x x x x x x (b) <- + = < = =- = - --- ), 1 ( 4 1 ), 1 ( 4 1 ) ( , ) , ( , ) , ( ) ( , 6 1 ) (...
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This note was uploaded on 04/08/2011 for the course MATH 3100 taught by Professor Mahdi during the Spring '11 term at University of the West Indies at Cave Hill.

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Ch3 - Ch.3 Homework Solution 3-5 (Buffons Neddle Problem) A...

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