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Confused on how the answer is .0372. Can someone please show work on how it is.

Confused on how the answer is .0372. Can someone please show work on how it is.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Confused on how the answer is .0372. Can someone please show work on how it is.
Example 3.22: Suppose that the shelf life, in years, of a certain perishable food product packaged in cardboard containers is a random variable whose probability density function is given by f(x) = -x e 0, 2 x > 0, elsewhere. Let X₁, X2, and X3 represent the shelf lives for three of these containers selected independently and find P(X₁ < 2,1 < X2 < 3, X3 > 2). Solution: Since the containers were selected independently, we can assume that the random variables X₁, X2, and X3 are statistically independent, having the joint probability density -2 f(x1, x2, 3) = f(x₁)ƒ(x₂)ƒ(x3) = e¯¹е-²е-³ for ₁ > 0, x2 > 0, x3 > 0, and f(x1, x2, 3) = elsewhere. Hence r8 P(X₁ < 2,1 < X2 <3, X3 > 2) = -1-2-13 = e e-1-2-3 dx₁ dx2 dx3 = (1-e-²) (e-¹e-³)e-² = 0.0372.
Transcribed Image Text:Example 3.22: Suppose that the shelf life, in years, of a certain perishable food product packaged in cardboard containers is a random variable whose probability density function is given by f(x) = -x e 0, 2 x > 0, elsewhere. Let X₁, X2, and X3 represent the shelf lives for three of these containers selected independently and find P(X₁ < 2,1 < X2 < 3, X3 > 2). Solution: Since the containers were selected independently, we can assume that the random variables X₁, X2, and X3 are statistically independent, having the joint probability density -2 f(x1, x2, 3) = f(x₁)ƒ(x₂)ƒ(x3) = e¯¹е-²е-³ for ₁ > 0, x2 > 0, x3 > 0, and f(x1, x2, 3) = elsewhere. Hence r8 P(X₁ < 2,1 < X2 <3, X3 > 2) = -1-2-13 = e e-1-2-3 dx₁ dx2 dx3 = (1-e-²) (e-¹e-³)e-² = 0.0372.
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