# Independence does not imply Zero Correlation

If I take $$X$$ to be a degenerate random variable, i.e. $$X=1$$ WP1 and $$Y=X$$ defined over the singleton sample space $$\Omega=\{1\}$$. Then

$$\mathbb{P}(X=1|Y=1)=1=\mathbb{P}(X=1)$$

i.e. I'd assume they're independent. But, we have

$$\rho_{XY}=\frac{\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)}{\sigma_X\sigma_Y}=\frac{0}{0}$$

which is undefined, not zero. Where am I being imprecise here/what's the misunderstanding? Thanks!

• Interesting question and a +1 from me. Notice that you get into similar trouble if you try to calculate the correlation between independent Cauchy-distributed random variables (not $0/0$, but a similar situation of not even being able to define the division).
– Dave
Mar 4, 2022 at 1:54
• You have not misunderstood. If either of the centered second moments are zero, then the Pearson correlation coefficient is indefinite. Now consider again if you had computed the reflective correlation coefficient instead. Mar 4, 2022 at 3:10

The point-mass example is worse, in a way. In the Cauchy example it's still true that the correlation is zero for, eg, all bounded functions $$f(X)$$, $$g(Y)$$ whereas in the point-mass example there's no way to get a well-defined correlation.