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suppose joint probability density function of $X,Y$ is $$f(x,y)=\frac{1}{2}\\ x>0 ,y>0,x+y<2$$how can i find probability density function $U=Y-X$

hadisanji
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    Letting $W$ be an independent Bernoulli$(1/2)$ variable, set $(\xi,\eta)=(1-X/2,Y/2)W + (Y/2,1-X/2)(1-W)$. Geometrically--draw a picture--it is clear that $\xi$ and $\eta$ are independent and have uniform$[0,1]$ distributions, while $U$ has the same distribution as $2(\xi+\eta)-2$. Therefore $U$ is the sum of two independent uniform variates, scaled by $2$ and shifted by $-2$. The PDF of $Z_2=\xi+\eta$ appears in the first plot of the question at http://stats.stackexchange.com/questions/41467 and the answers to that question provide many different ways to obtain the distribution of $U$. – whuber May 12 '14 at 14:17
  • You need the self-study tag. – Michael R. Chernick Aug 22 '19 at 22:34

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I will explain in words what you need to do as it seems a homework to me! Try to define another 1-1 transformation (say $V$) based on $X$ and $Y$. Then use the Jacobian theorem to find the joint density of $U$ and $V$. Finally, you can integrate this joint density to find the density function of $U$.

There is a similar example here that I suggest you to go over before doing this problem. Then this problem should a piece of cake for you to do!

Stat
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