Conditional variance

Conditional joint distribution’s Variance
Conditional variance of $X$ given that $Y = y$
- $Va r (X ∣ Y) = E [(X - E [X ∣ Y])^{2} ∣ Y] = E [X^{2} ∣ Y] - (E [X ∣ Y])^{2}$
- Hence $E [Va r (X ∣ Y)] = E [E [X^{2} ∣ Y]] - E [(E [X ∣ Y])^{2}] = E [X^{2}] - E [(E [X ∣ Y])^{2}]$
- Also, since $E [E [X ∣ Y]] = E [X]$ , we have $Va r (E [X ∣ Y]) = E [(E [X ∣ Y])^{2}] - (E [X])^{2}$
- Adding the last two equations, we have $Va r (X) = E [Va r (X ∣ Y)] + Va r (E [X ∣ Y])$

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