[Math] Two dependent random variables with standard normal distribution and zero covariance

Question

I need to find two dependent random variables with standard normal distribution, but with zero covariance. It is easy too find just two dependent random variables with such a distribution (X and -X, for example), but how I can reach zero covariance?

Thanks in advance.

Answer 1

Try $X \sim N(0,1)$ and $Y=X$ when $|X|\lt k$ and $Y=-X$ when $|X|\ge k$ for some non-negative $k$.

Then $X$ and $Y$ have standard normal distributions and $cov(X,Y)$ is a continuous increasing function of $k$, negative when $k$ is close to $0$ and positive when $k$ is large. So for some $k$ you will have $cov(X,Y)=0$