Let XX and YY be random variables. Then the conditional expectation

E[Y∣X] \mathbb{E} [Y \mid X]

is a random variable that depends on XX. Correspondingly we define its variance as

Var[E[Y∣X]]=E[(E[Y∣X])2]−(E[E[Y∣X]])2 \mathbb{Var} [ \mathbb{E} [Y \mid X]] = \mathbb{E} \left[ \left( \mathbb{E} [Y \mid X] \right)^2 \right] - ( \mathbb{E} \left[ \mathbb{E} [Y \mid X] \right] )^2

The Law of Total Variance states that

Var[Y]=E[Y2]−(E[Y])2=E[E[Y2∣X]]−E[(E[Y∣X])2]+E[(E[Y∣X])2]−(E[E[Y∣X]])2=E[E[Y2∣X]−(E[Y∣X])2]+Var[E[Y∣X]]=E[Var[Y∣X]]+Var[E[Y∣X]] \begin{aligned} \mathbb{Var} [Y] &= \mathbb{E} [Y^2] - \left( \mathbb{E} [Y] \right)^2 \\ &= \mathbb{E} [ \mathbb{E} [Y^2 \mid X]] - \mathbb{E} \left[ \left( \mathbb{E} [Y \mid X] \right)^2 \right] + \mathbb{E} \left[ \left( \mathbb{E} [Y \mid X] \right)^2 \right] - \left( \mathbb{E} \left[ \mathbb{E} [Y \mid X] \right] \right)^2 \\ &= \mathbb{E} \left[ \mathbb{E} [Y^2 \mid X] - \left( \mathbb{E} [Y \mid X] \right)^2 \right] + \mathbb{Var} [ \mathbb{E} [Y \mid X] ] \\ &= \mathbb{E} \left[ \mathbb{Var} [Y \mid X] \right] + \mathbb{Var} [ \mathbb{E} [Y \mid X] ] \end{aligned}

■\blacksquare