Answer to Question #180951 in Statistics and Probability for Sahil

Question #180951

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


1
Expert's answer
2021-04-29T17:52:54-0400

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\u2212\\xi,\\eta+\\xi)=cov(\\eta,\\eta+\\xi)\u2212cov(\\xi, \\eta+\\xi) =(cov(\\eta,\\eta)+cov(\\eta,\\xi))\u2212(cov(Y,\\eta)+cov(\\xi,\\xi))=var(\\eta)+cov(\\eta,\\xi)\u2212cov(\\xi,\\eta)\u2212var(\\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


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