Correlation of X and Y, when E[X] = 0

Question

X and Y are random variables and it is known that E(X) is zero. Then how can I prove that corr(X,Y)is zero?

Here is what I tried: The covariance between two random variables X and Y is defined as: Cov(X,Y)=E[(X−E[X])(Y−E[Y])] If E[X]=0

Then the covariance simplifies to: Cov(X,Y)=E[X(Y−E[Y])]=E[XY]−E[X]E[Y]=E[XY]−0⋅E[Y]=E[XY]

So, if E[X]=0, then E[XY]=E[X]⋅E[Y]=0⋅E[Y]=0

So the covariance is zero. Therefore, the correlation of X and Y is also zero. Does this hold for any two random variables (where one has zero expectation) or are there conditions which I have to assume. Or does the proof has some mistake?

Thanks

Answer 1

$$\mathbb{E}XY\ne\mathbb{E}X\mathbb{E}Y$$

In general, even when$\mathbb{E}X=0$ E.g. $$X=Y\sim N(0,1)$$ Then $$\mathbb{E}XY=\mathbb{E}X^2=1$$ And $$\mathbb{E}X=0$$