Suppose I want to randomly sample (x,y) from a bivariate gaussian/normal distribution:
$$f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y} \exp\left(-\frac12\left(\frac{x^2}{\sigma_x^2} + \frac{y^2}{\sigma_y^2}\right)\right)$$
Then is it a valid implementation of such random sampling method to separately sample x and y each from their own univariate gaussian distribution?
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma_x}{\exp({-\frac{1}{2}(\frac{x^2}{\sigma_x^2}}}))$; $\displaystyle f(y) = \frac{1}{\sqrt{2\pi}\sigma_y}{\exp({-\frac{1}{2}(\frac{y^2}{\sigma_y^2}}}))$
My intuition says yes because as I understand, if x and y are independent, then the combined probability is $P(x,y) = P(x)P(y)$. If not, then what is the recommended way to do this?