Two components of a minicomputer have the following joint pdf for their lifetimes X and Y: f(x,y)=xe−x(1+y), where x≥0,y≥0f(x,y)=0 otherwiseWhat is the probability that the lifetime X of the first component exceeds 3? What are the marginal pdf's of X and Y? Are the two lifetimes independent?

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Two components of a minicomputer have the following joint pdf for their lifetimes X and Y: f(x,y)=xe−x(1+y), where x≥0,y≥0f(x,y)=0 otherwiseWhat is the probability that the lifetime X of the first component exceeds 3? What are the marginal pdf&#039;s of X and Y? Are the two lifetimes independent?

Linda Tincher · Accepted Answer

The marginal pdf of X, by definition, should be (for x≥0)  f(x)=∫−∞∞f(x,y)dy=∫0∞xe−x(1+y)dy=−e−x∫0∞d(e−xy)=e−x  Thus, the probability that the life time X of the first component exceeds 3 is equal to  ∫3∞f(x)dx=∫3∞e−xdx=e−3  Similarly, the marginal pdf for Y should be (for y≥0 and using integration by parts)  ∫−∞∞f(x,y)dx=∫0∞xe−x(1+y)dx=∫0∞−x1+yd(e−x(1+y))=11+y2}  It is easy to check that f(x,y)≠f(x)f(y) in general, and hence the two life times are not independent.

Two components of a minicomputer have the following joint pdf for thei

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