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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) = 2e¹e-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). (c) Show that X₂ ~ Gamma(2, 1). (Recall that if X 2 ~ gamma(a, X), then fx (x) = = Aaa-¹e-Ax r(a) x > 0.)

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) = 2e¹e-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). (c) Show that X₂ ~ Gamma(2, 1). (Recall that if X 2 ~ gamma(a, X), then fx (x) = = Aaa-¹e-Ax r(a) x > 0.)

MATLAB: An Introduction with Applications
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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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). (c) Show that X₂ Gamma(2, 1). (Recall that if X~ gamma(a, X), then ƒx(x) = \ª¸ª-¹¸¯\ª x > 0.) I'(a)
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). (c) Show that X₂ Gamma(2, 1). (Recall that if X~ gamma(a, X), then ƒx(x) = \ª¸ª-¹¸¯\ª x > 0.) I'(a)
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