Function of a random vector

Summary

Definition

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

\[Z=g(X)\]

• is a new random variable.
• If \(X\) is an \(n×1\) random vector and \(g:R^n→R^m\) is a function, then

\[Y=g(X)\]

• is a new \(m×1\) random vector. In this case, we can write

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

• where each \(g_i:R^n→R\) for \(i=1,…,m\) . Thus

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

Linear functions of random vectors

• If \(g:R^n→R\) is a linear function, \(g\left( x \right)=c_1x_1+…c_nx_n=c'x\) where

\[c=\left( \right)\]

• is an \(n×1\) vector of constants, then

\[g\left( X \right)=c_1X_1+…+c_nX_n=\sum_{i}^{n}{ c_iX_i }=c'X\]

• 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

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

• is a \(m×1\) random vector.