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

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

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.
• Thank you. For the strict proof , I guess we may need to calculate marginal probabilities of z1 and z2 first, and verify the form $f(z1,z2) = g(z1)h(z2)$, then we can say they are independent . Aug 10 '19 at 8:30