Conditional Expectation and Variance

Properties:

Conditional Expectation

E(XY)E(X|Y)is the conditional expectation of XXgiven YY

  • E(XY=y)E(X|Y=y)is a fixed value, but E(XY)E(X|Y)is a random variable (it is a function of YY)

  • Iterated expectation: E(E(XY))=E(X)E(E(X|Y)) = E(X)

  • Additivity: E(Y+ZX)=E(YX)+E(ZX)E(Y+Z | X) = E(Y|X) + E(Z|X)

    • does not work on the right hand side: E(YX+Z)E(YX)+E(YZ)E(Y | X+Z) \ne E(Y|X) + E(Y|Z)

  • Linearity: E(aX+bY)=aE(XY)+bE(aX + b | Y) = aE(X|Y) + b

  • Conditioning on the same variable: E(g(S)TS)=g(S)E(TS)E(g(S)T | S) = g(S)E(T|S)

Conditional Variance

If Var(Y)Var(Y)is difficult to find directly, we can use the variance decomposition to condition the variance on another variable.

Var(Y)=E(Var(YX))+Var(E(YX))Var(Y) = E(Var(Y|X)) + Var(E(Y|X))