How can dependent events have a correlation of zero?

Question

I am working on a joint probability problem and am getting an answer I do not understand. {X: -1, -1, 0, 1, 2} Y = X^2

I determined that the events are dependent because P(X|Y) <> P(X) for all values of X and Y.

I then went on to identify the correlation co-efficient and got zero. But zero means there is no correlation. I would have expected that there would be a correlation for dependent events though.

Two questions:

1. Are they dependent or independent?
2. How can they be independent but have a correlation of zero?

I calculated the following ingredients:

x-bar=0, y-bar = 2, Var(X) = 2, Var(Y)=14/5, Cov(X,Y) = 0

Thanks

Answer 1

In answer to your questions.

1. They are dependent. You know exactly how they are dependent on each other through the defining equation $Y=X^2$.
2. If they are independent, they will have a correlation of zero: $$ \begin{align} Cov(X,Y) &= E(XY)-E(X)E(Y) \\ &= E(X)E(Y)-E(X)E(Y) \\ &= 0 \end{align} $$ but the converse is not true. You probably mean if they are dependent, how can they have a correlation of zero. The correlation you have computed measures the strength of linear relationship between the variables. Your variables are quadratically related and as you illustrated, for this example the linear correlation is zero.

Answer 2

Lack of correlation does not mean there is no relationship between the random variables. It means there is no linear relationship between the random variables.

Consequently, the correlation between $X$ and $Y$ may be zero even if they are dependent. Your computation illustrates just such an example.

Answer 3

We have $\operatorname{Cov}(X,Y) = E(X\cdot Y) - E(X)\cdot E(Y)$. If $X$ is a real valued random variable with a symmetric distribution about $0$ then $E(X) = \int x \rho(x) dx = 0$. For $Y = X^2$ we have $E(XY) = E(X^3) = \int x^3 \rho(x) d x = 0$, so $\operatorname{Cov}(X,X^2) = 0$, but $X$, $X^2$ are not independent.