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cookiesyum

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## Homework Statement

Let f(x

_{1}, x

_{2}, x

_{3}) = e

^{-(x1+x2+x3)}, 0<x

_{1,2,3}<infinity, zero elsewhere be a joint pdf of X

_{1}, X

_{2}, X

_{3}.

Compute P(X

_{1}< X

_{2}< X

_{3}) and P(X

_{1}= X

_{2}< X

_{3})

Determine the joint mgf.

## The Attempt at a Solution

P(X

_{1}< X

_{2}< X

_{3}) = triple integral of e

^{-(x1+x2+x3)}dx

_{2}dx

_{1}dx

_{3}as x

_{2}goes from x

_{1}to x

_{3}, x

_{1}goes from 0 to x

_{2}and x

_{3}goes from x

_{2}to infinity.

When I solve this integral I get 0 for an answer. The back of the book suggests the answer is 1/6. Clearly, I am setting up the problem wrong, but I don't know where my mistake is. Similarly, I'm having trouble setting up the second part of the question as well.

Joint mgf = E(e

^{t1x1+t2x2+t3x3}) but this integral is not easy by hand. Is there another way to do it?