User:Arthena/test

Sums of random variables and line integrals
The setting is: Let $$X$$ and $$Y$$ be independent random variables (on the reals), with density functions $$f_X$$ and $$f_Y$$. The standard result for the density of $$X+Y$$ is $$f_{X+Y}(a)=\int f_X(a-t)f_Y(t)dt$$. I'm trying to derive this formula by using a line integral, but I'm getting a different result, so I must be doing something wrong.

This is my reasoning: The joint density of $$(X,Y)$$ on $$R^2$$ is given by $$f_{X,Y}(a,b)=f_X(a)f_Y(b)$$. To find $$f_{X+Y}(a)$$ we should integrate $$f_{X,Y}$$ over the line $$x+y=a$$. We can parametrize this line by the function $$ r: R\rightarrow R^2, r(t)=(a-t, t) $$. Note that $$dr/dt=(-1,1)$$, so $$|dr/dt|=\sqrt{2}$$. Now if we calculate the line integral (as in Line_integral), we get $$f_{X+Y}(a)=\int f_{X,Y}(r(t))|dr/dt|dt=\int f_{X,Y}(a-t,t)\sqrt{2}dt=\int f_X(a-t)f_Y(t)\sqrt{2}dt$$. If we compare this with the textbook result above, then my result has a factor $$\sqrt{2}$$ that is not supposed to be there. Where did I go wrong? Arthena(talk) 10:03, 3 November 2010 (UTC)