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Conditional Expectation and Variance

Properties:

Conditional Expectation

​
E(X∣Y)E(X|Y)
is the conditional expectation of
XX
given
YY
​
  • ​
    E(X∣Y=y)E(X|Y=y)
    is a fixed value, but
    E(X∣Y)E(X|Y)
    is a random variable (it is a function of
    YY
    )
  • Iterated expectation:
    E(E(X∣Y))=E(X)E(E(X|Y)) = E(X)
    ​
  • Additivity:
    E(Y+Z∣X)=E(Y∣X)+E(Z∣X)E(Y+Z | X) = E(Y|X) + E(Z|X)
    ​
    • does not work on the right hand side:
      E(Y∣X+Z)≠E(Y∣X)+E(Y∣Z)E(Y | X+Z) \ne E(Y|X) + E(Y|Z)
      ​
  • Linearity:
    E(aX+b∣Y)=aE(X∣Y)+bE(aX + b | Y) = aE(X|Y) + b
    ​
  • Conditioning on the same variable:
    E(g(S)T∣S)=g(S)E(T∣S)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(Y∣X))+Var(E(Y∣X))Var(Y) = E(Var(Y|X)) + Var(E(Y|X))
​
Probability - Previous
The Gamma Family
Last modified 2yr ago