The lifetime of a certain component, in years, has probability density function.

Two such components, whose lifetimes are independent, are available. As soon as the first component fails, it is replaced with the second component. Let X denote the lifetime of the first component, and let Y denote the lifetime of the second component.

a. Find the joint probability density function of X and Y.

b. Find P(X ‰¤ 1 and Y > 1).

c. Find μ_X.

d. Find μ_X+Y.

e. Find P(X + Y ‰¤ 2). Sketch the region of the plane where x + y ‰¤ 2, and then integrate the joint probability density function over that region.

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