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

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

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