I'm currently learning about using the Dirac Delta and am working through a problem that I'm unsure about. Lets say I have two functions:
$$f(x) = \begin{cases} 1,& 0<x<1 \\ 0 ,& \text{else} \end{cases} \\ g(y) = \begin{cases} y, &0<y<1 \\ 0,& \text{else} \end{cases}$$
and define $z = x + y$. Then I want to see the distribution of $z$. If I want to determine this by solving
$h(z) = \int\int f(x) g(y) \delta(z - (x+y)) \,dx \,dy$ I first make a substitution $a = x+y$ Then $h(z) = \int\int f(a-y) g(a-x) \delta(z - a) \,da\, da = f(a)g(a) = 1(x+y).$ I'm not sure if I did this correct, I don't think it is because I'm not sure how to deal with the change of variables in this case.
I think, the integral with Dirac delta simply means give a value so $$\int \int dxdy f(x)g(y)\delta (z-(x+y))=\int dx f(x)g(z-x)=\int dy f(z-y)g(y)$$ I hope I'm not mistake and I hope this will help