CS70 Guide
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Conditional Expectation and Variance
Properties:

Conditional Expectation

E(XY)E(X|Y)
is the conditional expectation of
XX
given
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))