Problem: Let X and Y be independent random variables of the continuous type. Then X and.Y have the joint PDF: $f_{X,Y}(x,y)=f_X(x) f_Y(y)$. Prove that events ${a \leq X \leq b}$ and ${c \leq Y \leq d}$ are independent for any numbers $a,b,c,d$.
My answer: Since it has the joint PDF:$f_{X,Y}(x,y)=f_X(x) f_Y(y)$, and since X and Y are independent if and only if $f_{X,Y}(x,y)= f_X(x) f_Y(y)$, it is clear that $\int_{a}^b \int_{c}^d f_{X,Y}(x,y) dy dx=\int_{a}^b f_X(x) dx \int_{c}^d f_Y(y) dy $
Question: Can I leave like that?
That looks sufficiently convincing. Though it might be advisable to identify that those integrals do represent : $$\mathsf P(a\leqslant X \leqslant b\cap c\leqslant Y\leqslant d)=\mathsf P(a\leqslant X\leqslant b)~\mathsf P(c\leqslant Y\leqslant d)$$