1. Let X₁, X2, and X3 represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected values µ‚ µ², and µ and variances σ²,02, and o3, respectively a) If μ₁ = μ₂ = μ₂ = 60 and σ² = 0² = 03 =15, calculate P(X₁ + X₂ + X3 ≤ 200). b) Using the u's and o's given in part (a), calculate P(58 ≤ X ≤ 62), where X₁ + X₂ + X3 1 2 3 c) Using the u's and o's given in part (a), calculate P(-10 ≤X₁ – 0.5X₂ – 0.5X3 ≤5). d) If μ = 40, μ₂ = 50, µ = 60, o² =10, o² = 12, and o² =14, calculate P(X₁ + X₂ − 2X3 ≥ 0). X =
1. Let X₁, X2, and X3 represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected values µ‚ µ², and µ and variances σ²,02, and o3, respectively a) If μ₁ = μ₂ = μ₂ = 60 and σ² = 0² = 03 =15, calculate P(X₁ + X₂ + X3 ≤ 200). b) Using the u's and o's given in part (a), calculate P(58 ≤ X ≤ 62), where X₁ + X₂ + X3 1 2 3 c) Using the u's and o's given in part (a), calculate P(-10 ≤X₁ – 0.5X₂ – 0.5X3 ≤5). d) If μ = 40, μ₂ = 50, µ = 60, o² =10, o² = 12, and o² =14, calculate P(X₁ + X₂ − 2X3 ≥ 0). X =
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![1. Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \( \mu_1, \mu_2, \) and \( \mu_3 \) and variances \( \sigma_1^2, \sigma_2^2, \) and \( \sigma_3^2 \) respectively.
a) If \( \mu_1 = \mu_2 = \mu_3 = 60 \) and \( \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 15 \), calculate \( P(X_1 + X_2 + X_3 \leq 200) \).
b) Using the \( \mu\)'s and \( \sigma\)'s given in part (a), calculate \( P(58 \leq \bar{X} \leq 62) \), where
\[
\bar{X} = \frac{X_1 + X_2 + X_3}{3}.
\]
c) Using the \( \mu\)'s and \( \sigma\)'s given in part (a), calculate \( P(-10 \leq X_1 - 0.5X_2 - 0.5X_3 \leq 5) \).
d) If \( \mu_1 = 40, \mu_2 = 50, \mu_3 = 60, \sigma_1^2 = 10, \sigma_2^2 = 12, \) and \( \sigma_3^2 = 14 \), calculate \( P(X_1 + X_2 - 2X_3 \geq 0) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F121a245b-4265-470c-906a-aeb7dc29a223%2F73c46820-f0b3-4c55-b923-c2a322a8e52b%2Fs0do5q_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \( \mu_1, \mu_2, \) and \( \mu_3 \) and variances \( \sigma_1^2, \sigma_2^2, \) and \( \sigma_3^2 \) respectively.
a) If \( \mu_1 = \mu_2 = \mu_3 = 60 \) and \( \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 15 \), calculate \( P(X_1 + X_2 + X_3 \leq 200) \).
b) Using the \( \mu\)'s and \( \sigma\)'s given in part (a), calculate \( P(58 \leq \bar{X} \leq 62) \), where
\[
\bar{X} = \frac{X_1 + X_2 + X_3}{3}.
\]
c) Using the \( \mu\)'s and \( \sigma\)'s given in part (a), calculate \( P(-10 \leq X_1 - 0.5X_2 - 0.5X_3 \leq 5) \).
d) If \( \mu_1 = 40, \mu_2 = 50, \mu_3 = 60, \sigma_1^2 = 10, \sigma_2^2 = 12, \) and \( \sigma_3^2 = 14 \), calculate \( P(X_1 + X_2 - 2X_3 \geq 0) \).
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