Question

Let η and ξ be two independent normal random variables with mean 1 and variance 2. Which of the

following statements is correct?

◦ η + ξ and η − ξ are uncorrelated and independent

◦ η + ξ and η − ξ are uncorrelated, but not independent

◦ η + ξ and η − ξ are correlated, but independent

◦ η + ξ and η − ξ are correlated and not independent

◦ None of the statements is correct

Answer 1

η + ξ and η − ξ are independent only if η and ξ are normal random variables. Since it is exactly what is given in the question, η + ξ and η − ξ are independent.

$cov(\eta−\xi,\eta+\xi)=cov(\eta,\eta+\xi)−cov(\xi, \eta+\xi) =(cov(\eta,\eta)+cov(\eta,\xi))−(cov(Y,\eta)+cov(\xi,\xi))=var(\eta)+cov(\eta,\xi)−cov(\xi,\eta)−var(\xi)= var(\eta) - var(\xi)$

cov(X,Y) = cov(Y,X) =0 since X and Y are independent.

$cov(\eta + \xi, \eta - \xi) =var(\eta) - var(\xi) = 2-2=0$

Answer: η + ξ and η − ξ are uncorrelated and independent