How can I evaluate the marginal cumulative distribution function of a set of random variables for which I do not have the CDF in closed form. I can, however, simulate from a joint distribution involving this set of variables.
To be more specific, assume I want to evaluate the CDF of $(X_1,X_2)$ but I only have a way to simulate from $(X_1,X_2,X_3)$.
Obviously I can approximate the CDF of $(X_1,X_2,X_3)$ by obtaining a large number of simulations and checking how many observations fall below the desired threshold. But how to get the CDF of $(X_1,X_2)$?. Can I just simply throw $X_3$ away and use the same procedure as for the joint PDF?
Obviously I cant get $F_X(x) = P(X \leq x) = \lim_{y \to \infty} P(X \leq x, Y \leq y) = \lim_{y \to \infty} F_{XY} (x, y)$ because no closed form CDF is available. I also do not want to involve the pdf.