The expected value of a random vector

Summary

Definition

• \(X\) is an \(n×1\) random vector. We define

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

The expected value of a function of a random vector

• If \(X\) is an \(n×1\) continuous random vector with pdf \(f(x)\) and \(g:R^n→R\) is a function, then

\[E\left( g\left( X \right) \right)=\int_{R^n}{ g\left( x \right)f(x)dx }\]

• If \(X\) is an \(n×1\) continuous random vector with pdf \(f(x)\) and \(g:R^n→R^m\) is a function, then

\[g\left( x \right)=\left( \right)\]

• where \(g_j:R^n→R\) for \(j=1,…,m\) and

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

• is an \(m×1\) vector of constants where

\[E\left( g_j\left( X \right) \right)=\int_{R^n}{ g_j\left( x \right)f(x)dx } for j=1,…,m\]

Linear functions of random vectors

• If \(g:R^n→R\) is a linear function, \(g\left( x \right)=c'x\) where \(c\) is an \(n×1\) vector of constants then

\[E\left( c'X \right)=c'μ\]

• If \(g:R^n→R^m\) is a linear function, \(g\left( x \right)=Ax\) where \(A\) is an \(m×n\) matrix of constants then

\[E\left( AX \right)=Aμ\]