# CDF Approximation in Two Dimensions

Let $$(\Omega, \mathcal{A}, \mathbb{P})$$ be our respective probability space and $$X$$ a two-dimensional random variable with values on $$[0,1]^2$$ and probability density function $$f(x_1,x_2)$$ for $$x_1, x_2 \in [0,1]$$.

What are the most efficient methods (from numerics) to approximate the cumulative distribution function $$$$F(x, y) = \int_{[0,y]} \int_{[0,x]} f(x_1, x_2) \ dx_1 dx_2$$$$ for any $$x,y \in [0,1]$$ when a direct computation of the integral is not possible? Note I am interested in evaluating $$F(x,y)$$ for different values.

• The answer depends a lot on the degree of accuracy you are interested in, the smoothness of $f$, and how hard you want to work. – kimchi lover Mar 10 at 16:41
• I agree with the comment above by kimchi lover. This question in its current form is just too broad. Please elaborate your specific situation and needs. – Lee David Chung Lin Mar 12 at 0:38