recovering probability density from cumulative distribution

Question

Given a two-dimensional cumulative distribution

$$f(X<x,Y<y),$$ where $0<x,y<1$, how do I get a probability distribution of $x$, $P(x)$?

Is it simply $$P(x) = \frac{d}{dx}\int_0^1 dY\,f(X<x,Y<y)?$$

Answer 1

It might be more usual to start with the cumulative distribution function $$F_{X,Y}^{\,}(x,y)= P(X \le x, Y \le y)$$ and then say that the marginal density for $X$ is $$f_X^{\,}(x)=\frac d {dx}F_{X,Y}^{\,}(x,+\infty)$$

If instead you start with a density $f_{X,Y}^{\,}(x,y)$ where $F_{X,Y}^{\,}(x_0,y_0) =\int\limits_{-\infty}^{x_0}\int\limits_{-\infty}^{y_0} f_{X,Y}^{\,}(x,y) \, dy \,dx$ then you can say that the marginal density for $X$ is $$f_X^{\,}(x)=\int\limits_{-\infty}^{+\infty} f_{X,Y}^{\,}(x,y) \, dy$$