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Hw1Sol - Homework Solutions Assignment Number 1 Due Section...

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Unformatted text preview: Homework Solutions Assignment Number 1 Due January 22, 2007 Section 3.7 Problem 20. (a) • f X ( x ) = R ∞-∞ f X,Y ( x,y ) dy = R 2 x 1 2 dy = y 2 2 x = 1- x 2 , ≤ x ≤ 2 • f Y ( y ) = R ∞-∞ f X,Y ( x,y ) dx = R y 1 2 dx = x 2 y = y 2 , ≤ y ≤ 2 (b) • f X ( x ) = R ∞-∞ f X,Y ( x,y ) dy = R x 1 x dy = y x y = x y =0 = 1 , ≤ x ≤ 1 • f Y ( y ) = R ∞-∞ f X,Y ( x,y ) dx = R 1 y 1 x dx = ln | x || y =1 x = y =- ln y, ≤ y ≤ 1 (c) • f X ( x ) = R ∞-∞ f X,Y ( x,y ) dy = R 1- x 6 xdy = 6 xy | y =1- x y =0 = 6 x (1- x ) , ≤ x ≤ 1 • f Y ( y ) = R ∞-∞ f X,Y ( x,y ) dx = R 1- y 6 xdx = 3 x 2 1- y = 3(1- y ) 2 , ≤ y ≤ 1 Problem 23. p X,Y ( x,y ) = 4! x ! y !(4- x- y )! 1 2 x 1 3 y 1 6 4- x- y , ≤ x + y ≤ 4 The way to approach this problem is to try to just work the problem out directly from the definition: p X ( x ) = X all y p X,Y ( x,y ) = 4- x X y =0 4! x ! y !(4- x- y )! 1 2 x 1 3 y 1 6 4- x- y The problem here is that the value of x depends on y , since, for example, if y = 3, x can only be 0 or 1. There are two ways to handle this: The obvious way and the clever way. Probably the more obvious way is to just calculate the sum for every value. For example: p X (0) = ∑ all y p X,Y (0 ,y ) = ∑ 4 y =0 4!...
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Hw1Sol - Homework Solutions Assignment Number 1 Due Section...

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