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In a factory, there is a certain equipment which needs to be replaced immediately after it...

In a factory, there is a certain equipment which needs to be replaced immediately after it goes bad. Let X1 and X2 denote the first two times (in days) when the equipment is replaced. The joint density of X1 and X2 is given by fX1,X2 (x1, x2) = e −x2 , 0 ≤ x1 ≤ x2. (a) Compute the pdf of the lifetime of the equipment (i.e., the marginal pdf of X1). (b) Suppose that the second replacement happened after 15 days. Given that, what is the conditional distribution of the time of the first replacement? (Hint: compute fX1|X2 (x1|15).)

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