Deriving $Cov(X,Y,Z)$, is it even a thing?

So I am trying to derive a nice general formula for $Cov(X,Y,Z)$ and $Corr(X,Y,Z)$, I defined it as such $$Cov(X,Y,Z) = E[(X-E[X])(Y-E[Y])(Z-E[Z])]$$ $$Corr(X,Y,Z) = \frac {Cov(X,Y,Z)} {\sqrt{Var[X]Var[Y]Var[Z]}}$$

Now doing some algebra got me to this expression: $$Cov(X,Y,Z) = E[XYZ]-E[Z]E[XY]-E[X]E[ZY]-E[Y]E[XZ]+2E[X]E[Y]E[Z]$$

When I apply this formula however, my correlation is sometimes over 1 and below -1.

I have two questions:

1) Is the covariance ever defined this way (because I notice people always refer to the covariance matrix)?

2) Was my algebra wrong, or is the property $|COV|\le1$ is not transferable to 3 r.v.s?

• Notice that $E((X-E(X))^3) \neq E((X-E(X))^2)^{3/2} = \operatorname{Var}(X)^{3/2}$. Rather, it's something unrelated, called the third central moment of X. – aes Mar 13 '15 at 1:23
• so what you are saying is that this formulation of the Covariance is meaningless? – asosnovsky Mar 13 '15 at 1:25
• It isn't meaningless, but it isn't capturing some "single key relationship" between the three random variables like Correlation of two random variables does. Essentially it comes down to the fact that if you want to understand a function of three variables, you can't change variables such that there's one key number to pull out. What you've defined is one of the multivariate moments. – aes Mar 13 '15 at 1:31
• Oh okay, gotcha, thanks :D This is basically a generalized multivariate third moment. – asosnovsky Mar 13 '15 at 1:32