Can we say that 2 Gaussian random variables are independent if their bivariate normal joint density are standard?

Question

I know that , if $z_0$ and $z_1$ are 2 independent standard normal distributions (mean $u=0$ and variance $\sigma^2=1$) , we can have a bivariate standard normal distribution with the following pdf:

$f \left(z_0, z_1 \right) = \frac{1}{2 \pi } \exp \left\{ -\frac{1}{2} \left( z_0^2 + z_1^2 \right) \right\}, \ -\infty < z_0, \ z_1 <\infty$

I would like to ask that, given the pdf of above bivariate standard normal distribution, can we say $z_0$ and $z_1$ random variables are independent ?

Thank you

Answer 1

If the joint pdf of two random variables has the form $f(x,y)=g(x)h(y)$ then the random variables are independent. Hence the answer is YES.