Help with joint distributions?

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SUMMARY

The discussion centers on solving a joint distribution problem involving the joint density function f(x,y) = 2 for the region defined by 0 < y < x < 1. The correct probability P(X - Y > z) is derived as (1 - z)² / 2, as stated in the textbook. The initial attempt at the solution involved integrating the function incorrectly, leading to an erroneous result of z + 1/2. The user ultimately resolved their confusion and found the correct answer.

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Homework Statement


Suppose X and Y have joint density f(x,y)=2 for 0<y<x<1. Find P(X-Y>z)

According to the textbook the answer should be (1-z)^{2}/2

Homework Equations





The Attempt at a Solution



\int \int 2dxdy

for x=[0, z+y] and y=[0,1]

=\int 2(z+y) dy

=2z+1

since we are only interested in the values where y<x, we divide this by half to get

z+ 1/2

I'm clearly way off from the answer I'm suppose to get, so I would like to know where it is I went wrong. thanks!
 
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