The times in minutes X1,X2, and X3 to grade the 3 questions on a quiz are independent and normally distributed with means μ1= 6, μ2= 4, and μ3= 7, and standard deviations σ1= 1, σ2= 0.5, and σ3= 1.5. Let Y be the total time it takes to grade one such quiz. a) Find μY and σYb) Find the probability that Y is between 15 and 20 minutes

Answered: The times in minutes X1,X2, and X3 to…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The times in minutes X₁,X₂, and X₃to grade the 3 questions on a quiz are independent and normally distributed with means μ₁= 6, μ₂= 4, and μ₃= 7, and standard deviations σ₁= 1, σ₂= 0.5, and σ₃= 1.5. Let Y be the total time it takes to grade one such quiz.

a) Find μ_Y and σ_Y

b) Find the probability that Y is between 15 and 20 minutes

Expert Solution

Step 1

It is known that, X, Y and Z are three independent normal variables, such that, X ~ N(µ₁, σ₁²), Y ~ N(µ₂, σ₂²), and Z ~ N(µ₃, σ₃²), then X + Y + Z ~ N(µ₁ + µ₂ + µ₃, σ₁² + σ₂² + σ₃²),

Step 2

In this case, X₁ ~ N(6, 1²), X₂ ~ N(4, 0.5²), and X₃ ~ N(7, 1.5²),.

Thus,

Y = X₁ + X₂ + X₃ ~ N(6 + 4 + 7, 1² + 0.5² + 1.5²) ~ N(17, 3.5).

Thus, µ_Y = 17 and σ_Y = 1.8708 (= √3.5).

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