Confused about marginal density

Question

My book says the following

If you have two continuous random variables $X$ and $Y$ in a joint pdf $f(x,y)$ then

$f(y)$ = $\int_{-\infty}^\infty f(x,y)dx$

$f(x)$ = $\int_{-\infty}^\infty f(x,y)dy$

My question is, is this by definition or is there a proof or theorem that gives this result. I wish to know WHY this is true.

Answer 1

Notice how $$F_{X}(x)=\mathbb{P}(X\leq x)=\int_{-\infty}^{\infty} \int_{-\infty}^x f_{XY}(t,y)dtdy$$ Take a derivative with respective to $x$ gives

$$f_{X}(x)=\int_{-\infty}^{\infty}\frac{d}{dx}\Bigg[\int_{-\infty}^xf_{XY}(t,y)dt\Bigg]dy=\int_{-\infty}^{\infty}f_{XY}(x,y)dy$$ Similar for $f_{Y}(y)$.