Two dependent random variables with standard normal distribution and zero covariance 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.
 A: 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 $cov(X,Y)=0$
A: Let $X$ and $Z$ be independent standard normal random variables, and take $Y=Z$ on $\{X\cdot Z\ge 0\}$ and $Y=-Z$ on $\{X\cdot Z <0\}$. 
A: Although I haven't worked through all aspects that must be checked, I believe that if you specify
$$[X \mid Y] = Y$$
i.e. that $X$ becomes $Y$ when conditioning on $Y$, you obtain
$$\text {Cov}(X,Y) = E(XY) = E[E(XY\mid Y)] = E[YE(X\mid Y)]= E[YE(Y)] =0$$
Note that
$$E(X) = E[E(X\mid Y)] = E[Y] = 0$$
and by the decomposition-of-variance formula
$$\text {Var}(X) = \text {Var}(E[X\mid Y]) + E[\text {Var}(X\mid Y)]=  \text {Var}(E[Y])+E[\text {Var}(Y)] = 0 + 1 = 1$$
i.e. this conditional specification is consistent with the unconditional moments.
So these variables are uncorrelated but not independent.
