Conditional expectations and random vectors

Summary

Conditional expectation of a random variable

  • YY is a continuous random variable and XX is an n×1n×1 continuous random vector based on the same experiment.
  • We define the conditional expectation of YY given X=xX=x as

E(X=x)=∫∞−∞yfY|X(y|x)dyE(X=x)=∫∞−∞yfY|X(y|x)dy

  • E(X=x)E(X=x) is a function of xx which we may denote as g(x)g(x) .
  • The conditional expectation of YY given XX , E(X)E(X) , is then defined as the random variable g(X)g(X) .

Conditional expectation of a random vector

  • YY is an m×1m×1 continuous random vector and XX is an n×1n×1 continuous random vector based on the same experiment.
  • We define E(X)E(X) as an m×1m×1 random vector

E(X)=()E(X)=()

Results

  • If cc is an m×1m×1 vector of constants then

E(X)=c′E(X)

Var(X)=c′Var(X)c

  • which both are random variables.
  • If A is an k×m matrix of constants then

E(X)=AE(X)

Var(X)=AVar(X)A′

Total expectations and variances

E(E(X))=E(Y)

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