2 px 30000 f30000 00004 e 699 p20000 x

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Unformatted text preview: be operating. 5 9. 1 =1 λ a. E(X) = b. σ= c. P(X ≤ 4 ) = 1 − e d. P(2 ≤ X ≤ 5) = 1 − e a. P(X ≤ 100 ) = 1 − e 1 =1 λ −(1 )(4 ) = 1 − e −4 = .982 −(1 )(5 ) [ − 1 − e − (1)( 2) = e −2 − e −5 = .129 6 0. −(100 )(.01386) −( 200 )(.01386) = 1 − e −1. 386 = .7499 = 1 − e −2. 772 = .9375 P(X ≤ 200 ) = 1 − e P(100 ≤ X ≤ 200) = P(X ≤ 200 ) - P(X ≤ 100 ) = .9375 - .7499 = .1876 b. µ= 1 = 72.15 , σ = 72.15 .01386 P(X > µ + 2σ) = P(X > 72.15 + 2(72.15)) = P(X > 216.45) = [ 1 − 1 − e − ( 216.45)(.01386) = e −2.9999 = .0498 c. 6 1. ~ ~ ~ µ ) ⇒ 1 − e −( µ )(. 01386) = .5 ⇒ e − ( µ )(.01386) = .5 ~ ~ − µ (.01386) = ln(. 5) = .693 ⇒ µ = 50 .5 = P(X ≤ Mean = a. 1 = 25,000 implies λ = .00004 λ P(X > 20,000) = 1 – P(X ≤ 20,000) = 1 – F(20,000; .00004) − 1. 2 P(X ≤ 30,000) = F(30,000; .00004) = e = .699 P(20,000 ≤ X ≤ 30,000) = .699 - .551 = .148 b. σ= 1 = 25,000 , so P(X > µ + 2σ) = P( x > 75,000) = λ 1 – F(75,000;.00004) = .05. Similarly, P(X > µ + 3σ) = P( x > 100,000) = .018 150 = e −(. 00004)(20, 000) = .449 Chapter 4: Continuous Random Variables and Probability Distributions 6 9. µ=∫ ∞ 0 7 0. a. α α α −1 −( x β )α x αx α −1 x⋅ α x e dx = (after y = , dy = dx ) β β βα ∞1 1 −y β ∫ y α e dy = β ⋅ Γ1 + by definition of the gamma function. 0 α 2 ~ .5 = F ( µ ) = 1 − e −( µ / 3) ⇒ ~ ~ e − µ / 9 = .5 ⇒ µ 2 = −9 ln(. 5) = 6.2383 ⇒ µ = 2.50 b. 1 − e −[( µ− 3. 5) /1.5] = .5 ⇒...
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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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