Given a joint distribution function with this specific domain: $$f_{X,Y}(x,y) = g(x,y)*1_{x,y\ge 0,\space x+y\le1}$$
I'd like to find the expected values of $X,Y$, i'm just unsure about the integral bounds:
$$f_X(x)=\int_0^{1-x}f_{X,Y}(x,y)dy$$ $$f_Y(y)=\int_0^{1-y}f_{X,Y}(x,y)dx$$ $$E(X)=\int_0^{1-y}xf_{X}(x)dx$$ $$E(Y)=\int_0^{1-x}yf_{Y}(y)dy$$
Are the integral bounds correct? Or does the expected value bounds should be from 0 to 1?
The first two are correct. $EX$ and $EY$ are just numbers and they cannot involve variables. The correct formulas are $EX=\int_0^{1}xf_X(x)dx$ and $EX=\int_0^{1}yf_Y(y)dy$. $X$ and $Y$ take all values from $0$ to $1$.