Conditional variance

Definition:

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

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