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Ch3 - Ch.3 Homework Solution 3-5(Buffon’s Neddle Problem...

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Unformatted text preview: Ch.3 Homework Solution 3-5 (Buffon’s 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 π of . π Sol: Y : Y ~Uniform(0,D) : Θ Θ X H¼Ë » ª * ~Uniform(0,2 ) θ π ∫ ∫-- = = π θ θ π θ π 2 sin 2 sin 2 2 1 2 1 ) ( l D l D l dyd D x » ª * P P( ∴ & »ª* )= D l π 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 π 2 2 2 2 2 2 ∴ ab y x f XY π 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 π 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 ≤ ½). 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 = ∴ = ⋅- = Γ = =- = =- =- =- ≤ ≤- ∞ ≤ ≤- = ∫ ∫ ∫ ∫ ∫ ∫ ∞- ∞- ∞-- ∞--- 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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Ch3 - Ch.3 Homework Solution 3-5(Buffon’s Neddle Problem...

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