chapter 1

Chapter 4: Continuous Random Variables and Probability DistributionsSection 4.455.a. Γ(6) = 5! = 120⎛ 5 ⎞ 3 ⎛ 1 ⎞ 3 1 ⎛ 1 ⎞ ⎛ 3 ⎞b. Γ⎜⎟ = Γ⎜⎟ = ⋅ ⋅ Γ⎜⎟ = ⎜ ⎟ π ≈ 1. 329⎝ 2 ⎠ 2 ⎝ 2 ⎠ 2 2 ⎝ 2 ⎠ ⎝ 4 ⎠c. F(4;5) = .371 from row 4, column 5 of Table A.4d. F(5;4) = .735e. F(0;4) = P(X ≤ 0; α= 4) = 056.a. P(X ≤ 5) = F(5;7) = .238b. P(X < 5) = P(X ≤ 5) = .238c. P(X > 8) = 1 – P(X < 8) = 1 – F(8;7) = .313d. P( 3 ≤ X ≤ 8 ) = F(8;7) – F(3;7) = .653e. P( 3 < X < 8 ) =.653f. P(X < 4 or X > 6) = 1 – P(4 ≤ X ≤ 6 ) = 1 – [F(6;7) – F(4;7)] = .71357.a. µ = 20, σ 2 = 80 ⇒ αβ = 20, αβ 2 = 80 ⇒ β = 80 20, α = 5⎛ 24 ⎞b. P(X ≤ 24) = ⎜ ;5⎟ ⎝ 4 ⎠F = F(6;5) = .715c. P(20 ≤ X ≤ 40 ) = F(10;5) – F(5;5) = .41158. µ = 24, σ 2 = 144 ⇒ αβ = 24, αβ 2 = 144 ⇒ β = 6, α = 4a. P(12 ≤ X ≤ 24 ) = F(4;4) – F(2;4) = .424b. P(X ≤ 24 ) = F(4;4) = .567, so while the mean is 24, the median is less than 24. (P(X ≤µ ~ ) = .5); This is a result of the positive skew of the gamma distribution.149