Linear function of several random variables

Summary

Linear function

• \(X_1,…,X_n\) are \(n\) random variables and \(g:R^n→R\) is a linear function of \(n\) variables,

\[y=g\left( x_1,…,x_n \right)=a+b_1x_1+…+b_nx_n\]

• Then we say that

\[Y=g\left( X_1,…,X_n \right)=a+b_1X_1+…+b_nX_n\]

• is a linear function of the \(n\) random variables \(X_1,…,X_n\) .

Expected value of a linear function of \(n\) random variables

• If \(Y=a+b_1X_1+…+b_nX_n\) then

\[E\left( Y \right)=a+b_1E(X_1)+…+b_nE(X_n)\]

• or, combined,

\[E\left( a+b_1X_1+…+b_nX_n \right)=a+b_1E(X_1)+…+b_nE(X_n)\]

• ( \(E\) goes inside linear functions).

Variance of a linear function of \(n\) independent random variables

• If \(X_1,…,X_n\) are (mutually) independent then

\[Var\left( a+b_1X_1+…+b_nX_n \right)=b_1^2Var\left( X_1 \right)+…+b_n^2Var(X_n)\]

Example

• \(X_1,…,X_n\) are \(n\) independent random variables with \(E\left( X_i \right)=μ\) and \(Var\left( X_i \right)=σ^2\) for \(i=1,…,n\) . Then

\[E\left( X_1+…+X_n \right)=E\left( X_1 \right)+…+E\left( X_n \right)=nμ\]

\[Var\left( X_1+…+X_n \right)=Var\left( X_1 \right)+…+Var\left( X_n \right)=nσ^2\]