Ex5soln - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1306: INTRODUCTORY STATISTICS Sem 1 2008/2009 EXAMPLE CLASS 5 ANSWERS Q1. f ( x ) = ( ρ- 1) x- ρ for x ≥ 1 , ρ > 1 otherwise a) f(x) is a density function since Z ∞ 1 ( ρ- 1) x- ρ dx =- x- ( ρ- 1) ∞ 1 = 0- (- 1) = 1 b) F ( x ) = P [ X ≤ x ] = Z x-∞ f ( t ) dt = Z x 1 ( ρ- 1) t- ρ dt =- t- ( ρ- 1) x 1 = 1- x- ( ρ- 1) Therefore, F ( x ) = for x < 1 1- x- ( ρ- 1) for x ≥ 1 , ρ > 1 1 Q2. The p.d.f. of X is: f ( x ) = 1- | x- 1 | for 0 ≤ x ≤ 2 . (a) A sketch of f ( x )is: (b) The distribution function of X is: F ( x ) = Z x-∞ f ( t ) dt = for x < R x tdt = x 2 2 for ≤ x < 1 R 1 tdt + R x 1 (2- t ) dt = 1- (2- x ) 2 2 for 1 ≤ x < 2 1 for x ≥ 2 2 A sketch of the c.d.f. is: 3 Q3.Q3....
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Ex5soln - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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