Finding the conditional distribution from empirical joint CDF

Question

Suppose we are given a table containing the estimates of a discrete bivariate CDF $P(X_1\le x_1, X_2\le x_2)$ and also the known value of a realization of the first variable $X_1 = x_1$.

What are the steps for coming up with conditional distribution of $X_2$ given $X_1 = x_1$? (Is this procedure has a name, I can look up?)

Answer 1

$$P(X_2\le x_2 \mid X_1 = x_1) = \frac{P(X_2\le x_2 ,X_1 = x_1)}{P(X_1=x_1)}= \frac{P(X_2\le x_2 ,X_1 \le x_1)-P(X_2\le x_2 ,X_1 \le x_1-1)}{\lim_{x_2\to\infty} P(X_1\le x_1, X_2 \le x_2)-\lim_{x_2\to\infty} P(X_1\le x_1-1, X_2 \le x_2)}$$