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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

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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.

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  • $\begingroup$ 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 . $\endgroup$
    – zhfkt
    Commented Aug 10, 2019 at 8:30

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