F x f y dy 90 y 8 1 y dy 90 y 8 y 9

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Unformatted text preview: ± σ = (10.14, 17.36) Thus, P(µ - σ ≤ X ≤ µ + σ) = F(17.36) – F(10.14) = .5776 Similarly, P(µ - σ ≤ X ≤ µ + σ) = P(6.53 ≤ X ≤ 20.97) = 1 a. F(X) = 0 for x ≤ 0, = 1 for x ≥ 1, and for 0 < X < 1, 1 5. F( X ) = ∫ f ( y )dy = ∫ 90 y 8 (1 − y) dy = 90 ∫ ( y 8 − y 9 )dy x x −∞ ( 0 90 y − 1 9 x 9 1 10 10 y )] x 0 0 = 10 x − 9 x 9 10 F(x) 1.0 0.5 0.0 0.0 0.5 1.0 x b. F(.5) = 10(.5)9 – 9(.5)10 ≈ .0107 c. P(.25 ≤ X ≤ .5) = F(.5) – F(.25) ≈ .0107 – [10(.25)9 – 9(.25)10 ] ≈ .0107 – .0000 ≈ .0107 d. The 75th percentile is the value of x for which F(x) = .75 ⇒ .75 = 10(x)9 – 9(x)10 ⇒ x ≈ .9036 135 Chapter 4: Continuous Random Variables and Probability Distributions e. E(X) = ∫ ∞ x ⋅ f ( x )dx =∫ x ⋅ 90 x 8 (1 − x) dx = 90 ∫ x 9 (1 − x) dx 1 −∞ = 9x − 10 E(X2 ) = = ∫ ∞ x = 0 ≈ .8182 9 11 x 2 ⋅ f ( x )dx =∫ x 2 ⋅ 90 x 8 (1 − x )dx = 90 ∫ x 10 (1 − x) dx 1 −∞ 90 11 0 11 1 0 90 11 1 x− 11 90 12 12 1 0 x 1 0 0 ≈ .6818 V(X) ≈ .6818 – (.8182)2 = .0124, σx = .11134. f. µ ± σ = (.7068, .9295). Thus, P(µ - σ ≤ X ≤ µ + σ) = F(.9295) – F(.7068) = .8465 - .1602 = .6863 a. F(x) = 0 for x < 0 and F(x) = 1 for x > 2. For 0 ≤ x ≤ 2, 1 6. ∫ x y 2 dy = 1 y 3 8 3 08 F(x) = x 0 = 1 x3 8 F(x) 1.0 0.5 0.0 0 1 2 x () 113 82 = b. P(x ≤ .5) = F(.5) = c. P(.25 ≤ X ≤ .5) = F(.5) – F(.25) d. .75 = F(x) = e. E(X) = f. ∞ x ⋅ f ( x )dx =∫ x ⋅ 2 −∞ 2 ( ) x ⋅ 3 x 2 dx = 8 0 12 5 = 1 64 − 1 (1 ) = 84 3 7 512 ≈ .0137 x 3 ⇒ x3 = 6 ⇒ x ≈ 1.8171 ∫ ∫ E(X2 ) = V(X) = 1 8 1 64 − ( 3) = 2...
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This note was uploaded on 03/02/2014 for the course MATH 311 taught by Professor Yu during the Spring '08 term at Drexel.

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