Coskewness: three or two random variables? The wikipedia for coskewness says

coskewness is a measure of how much three random variables change
together

It then says

If two random variables exhibit positive coskewness they will tend to
undergo extreme positive deviations at the same time. Similarly, if
two random variables exhibit negative coskewness they will tend to
undergo extreme negative deviations at the same time.

Why is the definition for three random variables, yet the examples are able to explain it for only two random variables?
 A: I just noticed that, and edited the explanation on wikipedia you quoted, to reflect the three-variable definition used in the rest of the article. Though, looking around online, many people seem to be using "coskewness" to refer to a property of two random variables. Here's the relationship between the three-variable and two-variable definitions of coskewness: The three-variable definition is the one given on wikipedia: $S(X,Y,Z) = \frac{1}{\sigma_X\sigma_Y\sigma_Z}E[(X-\mu_X)(Y-\mu_Y)(Z-\mu_Z)]$, where $\mu_X$ and $\sigma_X$ refer to the mean and standard deviation, respectively, of $X$. The two-variable definition is just the same formula, but using the same variable for two of the inputs (i.e. $S(X,X,Y)$). I have no idea why anyone would want to restrict attention to this two-variable case, but it seems that it is popular to do so. Note that the three-variable definition of coskewness is symmetric between the three variables, but the two-variable definition of coskewness is not symmetric between its two variables, since $S(X,X,Y)\neq S(X,Y,Y)$. So any explanation of the two-variable notion of skewness that treats the two variables interchangably, including the one you quoted from wikipedia, is incorrect.
