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Probability: Difference between revisions
→Total Variance
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This one is not used as often on tests as total expectation | This one is not used as often on tests as total expectation | ||
{{hidden | Proof | | {{hidden | Proof | | ||
<math> | |||
\begin{aligned} | |||
Var(Y) &= E(Y^2) - E(Y)^2 \\ | |||
&= E(E(Y^2|X)) - E(E(Y|X))^2\\ | |||
&= E(Var(Y|X) + E(Y|X)^2) - E(E(Y|X))^2\\ | |||
&= E((Var(Y|X)) + E(E(Y|X)^2) - E(E(Y|X))^2\\ | |||
&= E((Var(Y|X)) + Var(E(Y|X))\\ | |||
\end{aligned} | |||
</math> | |||
}} | }} | ||