B3. A device has two components, and the lifetimes of these components are modelled by random variables Y₁ and Y₂. The first component cannot fail before the second component fails, and the joint density of Y₁ and Y₂ is determined to be f(y₁, y2) = 2e-¹e-92 for 0 < y2 < y₁ <∞. The random variable X₁ = Y₁ - Y₂ can be interpreted as the time between failures, and X₂ = Y₁+ Y₂ as the lifetime of the device. (a) Find the joint density of X₁ and X₂. (b) Show that X₁ Exponential (1).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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B3. A device has two components, and the lifetimes of these components are modelled by
random variables Y₁ and Y₂. The first component cannot fail before the second component
fails, and the joint density of Y₁ and Y₂ is determined to be
f(y₁, y2) 2e ¹e -92 for 0 < y2 < y1 <∞.
The random variable X₁ Y₁ - Y₂ can be interpreted as the time between failures, and
X₂ = Y₁ + Y₂ as the lifetime of the device.
=
(a) Find the joint density of X₁ and X₂.
(b) Show that X₁ Exponential (1).
Transcribed Image Text:B3. A device has two components, and the lifetimes of these components are modelled by random variables Y₁ and Y₂. The first component cannot fail before the second component fails, and the joint density of Y₁ and Y₂ is determined to be f(y₁, y2) 2e ¹e -92 for 0 < y2 < y1 <∞. The random variable X₁ Y₁ - Y₂ can be interpreted as the time between failures, and X₂ = Y₁ + Y₂ as the lifetime of the device. = (a) Find the joint density of X₁ and X₂. (b) Show that X₁ Exponential (1).
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