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This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics articles
This can't be right - the LHS of your proposed equation is measurable with respect to X_1, but the RHS is measurable with respect to X_2. This won't typically be the case. 130.88.123.107 (talk) 16:42, 3 December 2015 (UTC)[reply]
The variance is also known as the second central moment. Therefore the law of total variance could also be called the law of total second central moment.
Is there also such a thing as the law of total third central moment and the law of total fourth central moment?
I assume from the proof that the formula also holds for multivariate random variables. That is, if we replace Var with the Covariance-Matrix. It's not completely clear form the article right now though... --Thomasda (talk) 19:30, 9 February 2022 (UTC)[reply]
![{\displaystyle \operatorname {Var} [Y\mid X_{1}]=\operatorname {E} (\operatorname {Var} [Y\mid X_{1},X_{2}]\mid X_{1})+\operatorname {Var} (\operatorname {E} [Y\mid X_{1},X_{2}]\mid X_{1}).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a88d220e948bdaf8aee0b5dad0e8636ed36058e3)
![{\displaystyle \operatorname {Var} [Y\mid X_{1}]=\operatorname {E} (\operatorname {Var} [Y\mid X_{1},X_{2}]\mid X_{2})+\operatorname {Var} (\operatorname {E} [Y\mid X_{1},X_{2}]\mid X_{2}).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b501cab600e6440b44f419a975b2fed6209c870)