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

villane0

villane0

Answered question

2021-11-14

Two components of a minicomputer have the following joint pdf for their lifetimes X and Y: f(x,y)=xex(1+y), where x0,y0
f(x,y)=0 otherwise
What 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?

Answer & Explanation

Linda Tincher

Linda Tincher

Beginner2021-11-15Added 14 answers

The marginal pdf of X, by definition, should be (for x0)
f(x)=f(x,y)dy=0xex(1+y)dy=ex0d(exy)=ex
Thus, the probability that the life time X of the first component exceeds 3 is equal to
3f(x)dx=3exdx=e3
Similarly, the marginal pdf for Y should be (for y0 and using integration by parts)
f(x,y)dx=0xex(1+y)dx=0x1+yd(ex(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.

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