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Let (X1 ,X2, X3) be i.i.d. r.v.'s with common probability density function
f(x)= e^(-x), 0 <x and 0 otherwise
Find P(X1 <X3 -X2)
I'm having a difficult time setting up my integral
I know f(x1,x2,x3) = f(x1)*f(x2)*f(x3)= e^(-x1-x2-x3)
I can't figure out the limits.
Any help would certainly be appreciated
If the pdf was supported between zero and one, say, then the space would be a unit cube, and the region satsifying X1 < X3 - X2 ...
... or let's call it z < x - y ...
... would be the tetrahedron underneath z = x - y ...
http://www.ballooncalculus.org/draw/misc/tetra.png
Which would be...
... but extended becomes...
I really appreciate you breaking it down for me. That's what I ended up "guessing" but this was very helpful. When I solved the integral I got 1/4Quote:
Originally Posted by tom@ballooncalculus
If the pdf was supported between zero and one, say, then the space would be a unit cube, and the region satsifying X1 < X3 - X2 ...
... or let's call it z < x - y ...
... would be the tetrahedron underneath z = x - y ...
http://www.ballooncalculus.org/draw/misc/tetra.png
Which would be...
... but extended becomes...