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In the proof for the Law of Total Variance, the following lemma seems to be appealed to (when going from the 2nd to the 3rd step of the proof):

$$ E[E[Y^2 \mid X]] = E[\text{Var}[Y \mid X] + [E[Y \mid X]]^2] $$

Where does this come from and/or what justifies it?

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This is the expectation of

$$ E[Y^2\mid X]=\operatorname{Var}[Y\mid X]+E[Y\mid X]^2\;, $$

which is the conditional version of the definition of the variance,

$$ E[Y^2]=\operatorname{Var}[Y]+E[Y]^2\;. $$

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Discrete version: $$ \text{Var}[Y|X]=\sum_{y}(y-E[Y|X])^2p_{Y|X}(y|x)\\=\sum_yy^2p_{Y|X}(y|x)-2E[Y|X]\sum_y yp_{Y|X}(y|x)+E[Y|X]^2\sum_{y}p_{Y|X}(y|x)\\ =E[Y^2|X]-2E[Y|X]^2+E[Y|X]^2=E[Y^2|X]-E[Y|X]^2\\ \therefore E[Y^2|X]=\text{Var}[Y|X]+E[Y|X]^2 $$

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