# hw3sol - F x = 0 25 ⇒ the lower quartile is 4.43(d...

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HW3 Solution Section 2.3 1. EX = 0 × 0 . 08 + 1 × 0 . 11 + 2 × 0 . 27 + 3 × 0 . 33 + 4 × 0 . 21 = 2 . 48 10. (a) EX = R 6 4 x 1 x ln 1 . 5 dx = 2 ln 1 . 5 = 4 . 93 (b) Since F ( x ) = 1 ln 1 . 5 ln x 4 , solve x 0 for F ( x 0 ) = 0 . 5 1 ln 1 . 5 ln x 0 4 = 0 . 5 x 0 = 4 . 90 11. (a) EX = R 4 0 x 1 8 xdx = 1 8 x 3 3 ± ± ± 4 0 = 2 . 67 (b) F ( x 0 ) = x 2 0 16 = 0 . 5 x 0 = 2 . 83 Section 2.4 1. (Check solution of Textbook.) (a) EX = 1 3 ( - 2) + 1 6 (1) + 1 3 (4) + 1 6 (6) = 11 6 . (b) V arX = 1 3 ( - 2 - 11 6 ) 2 + 1 6 (1 - 11 6 ) 2 + 1 3 (4 - 11 6 ) 2 + 1 6 (6 - 11 6 ) 2 = 9 . 47 (c) EX 2 = 1 3 ( - 2) 2 + 1 6 (1) 2 + 1 3 (4) 2 + 1 6 (6) 2 = 12 . 83 . V arX = EX 2 - ( EX ) 2 = 9 . 47 5. (a) EX = 4 . 93, EX 2 = Z 5 4 x 2 1 x ln(1 . 5) dx = 1 ln(1 . 5) 1 2 x 2 ± ± ± ± 6 4 = 24 . 66 V arX = EX 2 - ( EX ) 2 = 24 . 66 - 4 . 93 2 = . 3551 (Note, if you use EX = 4 . 94, V arX = 0 . 25, either answer is Fne). (b) σ = V arX = . 3551 = 0 . 6 (c) The cdf is F ( x ) = 1 ln(1 . 5) (ln x - ln 4) when 4 x 6. The upper quartile is to solve x for F ( x ) = 0 . 75 the upper quartile is 5.42. The lower quartile is to solve x for
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Unformatted text preview: F ( x ) = 0 . 25 ⇒ the lower quartile is 4.43. (d) Interquartile range = 5 . 42-4 . 43 = 0 . 99. 6. (a) The pdf is f ( x ) = 1 8 x . EX = 8 3 , EX 2 = 8 , V arX = 8 9 . (b) σ = √ V arX = q 8 9 = . 94. (c) The upper quartile is to solve x for F ( x ) = 0 . 75 ⇒ the upper quartile is 3.46. The lower quartile is to solve x for F ( x ) = 0 . 25 ⇒ the lower quartile is 2. (d) Interquartile range = 3 . 46-2 = 1 . 46. 1...
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