Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values μ₁, μ2, and μ3 and variances σ₁ ЛUSE SALT (a) If μ₁ = μ₂ = M3 = 60 and σ₁ 2 2 = σ = 18, calculate P(T ≤ 204) and P(144 ≤ T ≤ 204). P(T。 ≤ 204) = 0.9995 P(144 ≤ To ≤ 204) = 0.9995 (b) Using the μ,'s and σ,'s given in part (a), calculate both P(54 ≤ X) and P(58 ≤ X ≤ 62). P(54 ≤ X) = 0.9214 P(58X62) = 0.5858 (c) Using the μ's and σ's given in part (a), calculate P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6). - P(-12 ≤ X₁ = 0.5X2 - 0.5X3 ≤ 6) = 0 Interpret the quantity P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6). ○ The quantity represents the probability that the difference between X3 and the average of X₁ and X2 is between -12 and 6. The quantity represents the probability that X1, X2, and X3 are all between -12 and 6. The quantity represents the probability that the difference between X₁ and the average of X2 and X3 is between -12 and 6. O The quantity represents the probability that the difference between X3 and the sum of X1 and X2 is between -12 and 6. O The quantity represents the probability that the difference between X and the sum of X2 and X3 is between -12 and 6. (d) If μ₁ = 50, M2 = 60,3 = 70, 10,02² P(X1 + x2 + X3 ≤ 190) P(X1 + x2 ≥ 2X3) 2 = 0.6832 = 0.1448 × 2 03 = 14, and σ3² = 12, calculate P(X₁ + X 2 + X3 ≤ 190) and also P(X₁ + X₂ ≥ 2X3). 2 02 and σ32, respectively. (Round your answers to four decimal places.)
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values μ₁, μ2, and μ3 and variances σ₁ ЛUSE SALT (a) If μ₁ = μ₂ = M3 = 60 and σ₁ 2 2 = σ = 18, calculate P(T ≤ 204) and P(144 ≤ T ≤ 204). P(T。 ≤ 204) = 0.9995 P(144 ≤ To ≤ 204) = 0.9995 (b) Using the μ,'s and σ,'s given in part (a), calculate both P(54 ≤ X) and P(58 ≤ X ≤ 62). P(54 ≤ X) = 0.9214 P(58X62) = 0.5858 (c) Using the μ's and σ's given in part (a), calculate P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6). - P(-12 ≤ X₁ = 0.5X2 - 0.5X3 ≤ 6) = 0 Interpret the quantity P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6). ○ The quantity represents the probability that the difference between X3 and the average of X₁ and X2 is between -12 and 6. The quantity represents the probability that X1, X2, and X3 are all between -12 and 6. The quantity represents the probability that the difference between X₁ and the average of X2 and X3 is between -12 and 6. O The quantity represents the probability that the difference between X3 and the sum of X1 and X2 is between -12 and 6. O The quantity represents the probability that the difference between X and the sum of X2 and X3 is between -12 and 6. (d) If μ₁ = 50, M2 = 60,3 = 70, 10,02² P(X1 + x2 + X3 ≤ 190) P(X1 + x2 ≥ 2X3) 2 = 0.6832 = 0.1448 × 2 03 = 14, and σ3² = 12, calculate P(X₁ + X 2 + X3 ≤ 190) and also P(X₁ + X₂ ≥ 2X3). 2 02 and σ32, respectively. (Round your answers to four decimal places.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values μ₁, μ2, and μ3 and variances σ₁
ЛUSE SALT
(a) If μ₁ = μ₂ = M3 = 60 and σ₁ 2
2
= σ
= 18, calculate P(T ≤ 204) and P(144 ≤ T ≤ 204).
P(T。 ≤ 204)
= 0.9995
P(144 ≤ To ≤ 204)
= 0.9995
(b) Using the μ,'s and σ,'s given in part (a), calculate both P(54 ≤ X) and P(58 ≤ X ≤ 62).
P(54 ≤ X) = 0.9214
P(58X62) = 0.5858
(c) Using the μ's and σ's given in part (a), calculate P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6).
-
P(-12 ≤ X₁ = 0.5X2 - 0.5X3 ≤ 6) = 0
Interpret the quantity P(-12 ≤ X₁ - 0.5X2 - 0.5X3 ≤ 6).
○ The quantity represents the probability that the difference between X3 and the average of X₁ and X2 is between -12
and 6.
The quantity represents the probability that X1, X2, and X3 are all between -12 and 6.
The quantity represents the probability that the difference between X₁ and the average of X2 and X3 is between -12
and 6.
O The quantity represents the probability that the difference between X3 and the sum of X1 and X2 is between -12 and 6.
O The quantity represents the probability that the difference between X and the sum of X2 and X3 is between -12 and 6.
(d) If μ₁ = 50, M2
= 60,3 = 70, 10,02²
P(X1 + x2 + X3 ≤ 190)
P(X1 + x2 ≥ 2X3)
2
= 0.6832
= 0.1448
×
2
03
= 14, and σ3² = 12, calculate P(X₁ + X 2 + X3 ≤ 190) and also P(X₁ + X₂ ≥ 2X3).
2
02
and σ32, respectively. (Round your answers to four decimal places.)
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