# Two uncorrelated random variables are not necessarily independent

We have seen that if two random variables \(X\) and \(Y\) are independent, then their covariance is zero,

\[
\mathbf{C}[X,Y] = 0,
\]

and therefore their correlation coefficient is also zero:

\[
\rho(X,Y) = 0.
\]

Does the reverse hold?
Namely, if you find that the correlation between two random variables is zero, does this imply that they are independent?
The answer to this question is a loud **NO**.
We will show that it does not hold through a counter example.

Take these two independent random variables:

\[
X \sim N(0, 1),
\]

and

\[
Z \sim N(0, 1).
\]

Then define this new random variable \(Y\) by:

\[
Y = X^2 + 0.1 Z.
\]

Since there is a functional relationship between \(X\) and \(Y\), they are obviously not independent.
But let’s generate some data from them and estimate the correlation:

It’s instructive to look at the scatter plot:

Well, it’s obvious that they are not independent.
Let’s see what the correlation coefficient is:

Very close to zero.
So, \(X\) and \(Y\) are uncorrelated…
Rememeber this please!
Do the scatter plots and use your common sense.
Do not just rely on a number to make decisions.

After you see the scatter plot like this, you get suspicous.
You start thinking that there may be a correlation between the square of \(X\) and \(Y\).
Let’s estimate the correlation of \(X^2\) and \(Y\) to see what it turns out to be:

Almost one! (It is actually exactly one).