Sampling from bivariate joint cumulative distribution function Given two variable $x,y$, they are subjected to a joint probability density function:
$
f(x,y) = \dfrac{1}{3}(3x^2 + 4xy + 3y^2)\\
0\leq x \leq 1;0\leq y \leq 1
$
Obviously, its corresponding cumulative distribution function (CDF) is:
$
F(x,y) = \dfrac{1}{3}(x^3y+x^2y^2+xy^3)
$
My objective is to generate random samples from this CDF using the inverse tranformation method.
As far as I know, the first step is to get the marginal CDF of $x$:
$
F_x = F(x,y=1) = \dfrac{1}{3}(x^3 + x^2 + x)
$
Then we can readily get a random value of $x$, denoted as $u_x$.
Next, for sampling $y$, we have to derive its conditional CDF $F_{y|x}$ given $x$.
My question is how to derive the $F_{y|x}$? Can the well-known conditional rule still be adapted for this problem, such that:
$
F_{y|x} = \dfrac{F(x,y)}{F_x}
$
It is worth noting that the actual CDF I encountered is much more complicated than the above example, and is very difficult to perform integration on the PDF or CDF.
Thank you very much!

If$$f(x,y) = \dfrac{1}{3}(3x^2 + 4xy + 3y^2)$$
then $$f_{Y|X=x}(y)\propto f(x,y)\propto 3x^2 + 4xy + 3y^2$$
leads to$$f_{Y|X=x}(y)=\dfrac{3x^2 + 4xy + 3y^2}{\underbrace{\int_0^1 (3x^2 + 4xy + 3y^2)\,\text dy}_{3x^2+2x+1}}$$
from which the conditional cdf can be derived.
