# What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?

What is the meaning of the quantity:

$$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$

Is there, for example, a geometric explanation? Is there a term for it in statistics?

• if the answer ok for you @apadana, can you please accept and/or upvote it? Mar 26 '20 at 20:50

It is the square root of the determinant of the covariance matrix (between $$X$$ and $$P$$). The determinant of the covariance matrix is called as Generalized Variance, which quantifies the co-variability of multivariate random variables to a scalar. What you write is the square root of it, so I believe it won't be too odd to call it as Generalized Deviation.
• "Co-variability" is more accurate, than "variability". E.g. if the determinant of a covariance matrix is zero, there's linear dependence among the de-meaned random variables. (In the two by two case here, $\epsilon = 0$ if and only if $X$ lies in the linear span of $P$ and $1$.) Mar 12 '20 at 1:28