Conditional expectations and random vectors

Summary

Conditional expectation of a random variable

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

\[E\left( X=x \right)=\int_{-∞}^{∞}{ yf_{Y|X}\left( y|x \right)dy }\]

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

Conditional expectation of a random vector

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

\[E\left( X \right)=\left( \right)\]

Results

• If \(c\) is an \(m×1\) vector of constants then

\[E\left( X \right)=c'E\left( X \right)\]

\[Var\left( X \right)=c'Var\left( X \right)c\]

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

\[E\left( X \right)=AE\left( X \right)\]

\[Var\left( X \right)=AVar\left( X \right)A'\]

Total expectations and variances

\[E\left( E\left( X \right) \right)=E\left( Y \right)\]

\[Var\left( Y \right)=E\left( Var\left( X \right) \right)+Var\left( E\left( X \right) \right)\]