Discussion7

ContinuoUs random variables

Problem 09 page 279

Let \(X\), \(Y\) and \(Z\) be independent random variables each with mean \(\mu\) and variance \(\sigma^{2}\) a) Find \(E(S)\) and \(V(S)\) where \(S=X+Y+Z\) b) Find \(E(A)\) and \(V(A)\) where \(A=\frac{1}{3}(X+Y+Z)\) c) Find \(E(S^{2})\) and \(E(A^{2})\)

Solution:

\(E(S)=E(X+Y+Z)=E(X)+E(Y)+E(Z)=3\mu\)

\(V(S)=V(X+Y+Z)=V(X)+V(Y)+V(Z)=3\sigma^{2}\)

\(E(A)=E(\frac{1}{3}(X+Y+Z))=\frac{1}{3}E(S)=\mu\)

\(V(A)=\frac{1}{3^2}V(S)=\frac{\sigma^{2}}{3}\)

\(E(S^2)=V(S)+(E(S))^2=3\sigma^{2}+(3\mu)^2=3\sigma^{2}+9\mu^2\)

\(E(A^2)=V(A)+(E(A))^2=\frac{\sigma^{2}}{3}+\mu^2\)