Let $Y = (Y_1,Y_2)$ be a random vector with denisty $$f(y_1,y_2) = \frac{2}{\pi} e^{-\frac{y^2_1+y^2_2}{2}} I(y_1 >0,y_2>0)$$
Are $Y_1$ and $Y_2$ independent?
Two variables are independent if $f(x,y) = f(x)f(y)$
Then could I make the argument that they are independent since $$f(y_1) = \frac{1}{\pi}e^{\frac{-Y^{2}_1}{2}}$$ and $$f(y_2) = 2e^{\frac{-Y^{2}_2}{2}}$$
It is easy to verify that the join distribution can be factorized as follows
$$f_{XY}(x,y)=\frac{2}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\mathbb{1}_{(0;+\infty)}(x)\times\frac{2}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}\mathbb{1}_{(0;+\infty)}(y)=f_X(x)f_Y(y)$$
Thus X,Y are independent, both disributed as a conditional Standard Gaussian (conditioned on $X>0$,$Y>0$)
You can get the same result integrating the opposite variable to find a marginal density