Law of Total Variance — $\operatorname{Var}(X)$ = ?

hard

Answer

$\operatorname{Var}(X)\; = \;E[\operatorname{Var}(X\; \mid \;Y)]\; + \;\operatorname{Var}(E[X\; \mid \;Y])$
$'Within-group' variance (average of conditional variances) plus 'between-group' variance (variance of conditional means).$
$(R^{2}\; = \;\mathrm{explained} / \mathrm{total})$

Check yourself — multiple choice

$\operatorname{Var}(X)\; = \;E[X]^{2}$
$\operatorname{Var}(X)\; = \;E[\operatorname{Var}(X \mid Y)]\; + \;\operatorname{Var}(E[X \mid Y])$ — within + between decomposition; foundation of $\mathrm{ANOVA}\; / \;R^{2}$
$\operatorname{Var}(X)\; = \;\operatorname{Var}(Y)$
Never used

Law of total variance: within + between decomposition; ANOVA, $R^{2}$ , mixed effects.

#variance#probability

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