IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables
For random variables X and Y with finite variances, which statement about conditional expectation is correct?
The correct statement is that Var(X) equals E[Var(X|Y)] plus Var(E[X|Y]). This is the law of total variance, splitting variability into the average within-group part and the between-group part.
- AE[E[X|Y]] equals E[Y]
- BVar(X) equals E[Var(X|Y)] plus Var(E[X|Y])Correct
- CVar(X) equals Var(E[X|Y]) minus E[Var(X|Y)]
- DE[X|Y] is always a constant if X and Y are uncorrelated
- Var(X) is never less than E[Var(X|Y)] only when X and Y are independent
Explanation
The law of total variance states Var(X)=E[Var(X|Y)]+Var(E[X|Y]). E[E[X|Y]] equals E[X], not E[Y]. Uncorrelated does not imply E[X|Y] is constant, and Var(X) is always at least E[Var(X|Y)].
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