# soln3 - Soln3-1 SS-1 Soln3-2 1 Soln3-3 Soln3-4 Soln3-5 2...

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Unformatted text preview: Soln3-1 SS-1. Soln3-2 1. Soln3-3 Soln3-4 Soln3-5 2. Soln3-6 Soln3-7 SS-2. Soln3-8 SS-3. Soln3-9 SS-4. Soln3-10 SS-5. (a) Z ∞- ∞ f X ( x ) dx = 1 ⇒ Area under triangle of height c and base 2 a equals 1 ⇒ 1 2 ( c )( 2 a ) = 1 ⇒ c = 1 / a The f X ( x ) = 1 a- | x | a 2 , | x | ≤ a , otherwise (b) For x <- a , F X ( x ) = For- a ≤ x < 0, F X ( x ) = Z x- a f X ( x ) dx Soln3-11 equals the area of triangle of base x- (- a ) and height x + a a 1 a = x + a a 2 (using similar triangles) Thus, F X ( x ) = ( x + a ) 2 2 a 2 ,- a ≤ x < For 0 ≤ x < a , F X ( x ) = 1- Z a x f X ( x ) dx where the integral equals the area of triangle of base a- x and height a- x a 1 a = a- x a 2 (using similar triangles) Thus, F X ( x ) = 1- ( a- x ) 2 2 a 2 , ≤ x < a For x > a , F X ( x ) = 1 (c) By symmetry, P [ | X | < b ] = 1 / 2 ⇒ P [ < X < b ] = 1 / 4. The probability is 1/2 the area under a triangle of base a- b and height a- b a 1 a = a- b a 2 Thus, the probability is 1 2- ( a- b ) 2 2 a...
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soln3 - Soln3-1 SS-1 Soln3-2 1 Soln3-3 Soln3-4 Soln3-5 2...

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