Linear function of several random variables

Summary

Linear function

  • X1,…,XnX1,…,Xn are nn random variables and g:Rn→Rg:Rn→R is a linear function of nn variables,

y=g(x1,…,xn)=a+b1x1+…+bnxny=g(x1,…,xn)=a+b1x1+…+bnxn

  • Then we say that

Y=g(X1,…,Xn)=a+b1X1+…+bnXnY=g(X1,…,Xn)=a+b1X1+…+bnXn

  • is a linear function of the nn random variables X1,…,XnX1,…,Xn .

Expected value of a linear function of nn random variables

  • If Y=a+b1X1+…+bnXnY=a+b1X1+…+bnXn then

E(Y)=a+b1E(X1)+…+bnE(Xn)E(Y)=a+b1E(X1)+…+bnE(Xn)

  • or, combined,

E(a+b1X1+…+bnXn)=a+b1E(X1)+…+bnE(Xn)E(a+b1X1+…+bnXn)=a+b1E(X1)+…+bnE(Xn)

  • ( EE goes inside linear functions).

Variance of a linear function of nn independent random variables

  • If X1,…,XnX1,…,Xn are (mutually) independent then

Var(a+b1X1+…+bnXn)=b21Var(X1)+…+b2nVar(Xn)Var(a+b1X1+…+bnXn)=b21Var(X1)+…+b2nVar(Xn)

Example

  • X1,…,XnX1,…,Xn are nn independent random variables with E(Xi)=μE(Xi)=μ and Var(Xi)=σ2Var(Xi)=σ2 for i=1,…,ni=1,…,n . Then

E(X1+…+Xn)=E(X1)+…+E(Xn)=nμE(X1+…+Xn)=E(X1)+…+E(Xn)=nμ

Var(X1+…+Xn)=Var(X1)+…+Var(Xn)=nσ2Var(X1+…+Xn)=Var(X1)+…+Var(Xn)=nσ2