Show a univariate random variable is independent of Bivariate random variable

Question

Suppose we have two normal random variables $X$ and $Y$. To show $X$ and $Y$ are independent, one must show: $\\ Cov(X,Y) = E(XY) - E(X)E(Y) = 0$

Now suppose $X$ is a univariate normal random variable, and $Y = [Y_1, Y_2]^T$ where $Y_1,Y_2$ are also normal random variables.

Does showing the $X$ is independent from $Y_1$ and $Y_2$ separately, mean $X$ is independent from $Y$?

i.e:

$\\ Cov(X,Y_1) = E(XY_1) - E(X)E(Y_1) = 0$

$\\ Cov(X,Y_2) = E(XY_2) - E(X)E(Y_2) = 0$

implies $X \perp Y$

Answer 1

Your initial premise is wrong. Zero covariance shows that the random variables are uncorrelated (that is, not linearly correlated). They might nevertheless be dependent. For instance, if $X$ is uniformly distributed on $\{-1,0,1\}$ and $Y=|X|$, then $X$ and $Y$ are uncorrelated but dependent.