The law of the total third central moment

Question

On Wikipedia you can read in the article: The law of total variance as follows.

In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if $X$ and $Y$ are random variables on the same probability space, and the variance of $Y$ is finite, then

$$\displaystyle {\it Var} \left( Y \right) =E \left( {\it Var} \left( Y | X \right) \right) +{\it Var} \left( E \left( Y | X \right) \right)$$

The variance is also called the second central moment. Therefore, the law of total variance could also be called the law of the total second central moment. Is there also such a thing as the law of the total third central moment and the law of the total fourth central moment?

Answer 1

It is a special case of the Law of total cumulance (see Wikipedia). For the third central moment one obtains:

$\mu_3(X)= \operatorname{E}(\mu_3(X\mid Y))+\mu_3(\operatorname{E}(X\mid Y)) +3\operatorname{cov}(\operatorname{E}(X\mid Y),\operatorname{var}(X\mid Y)).$