1) Let X and Y be in independent standard normal random variables, and define a new rv by
U = .6X + .8Y.
a) Determine Corr(X, U).
b) How would you alter U to obtain Corr(X, U) = p for a specified value of p?
Please help me out with this problem
Please clearly show me how to do this problem
Thank you so much
U = .6X + .8Y.
a) Determine Corr(X, U).
b) How would you alter U to obtain Corr(X, U) = p for a specified value of p?
Please help me out with this problem
Please clearly show me how to do this problem
Thank you so much
-
a)
corr(X,U) = cov(X,U)/sqrt(var(X)var(U))
cov(X,U)=cov(X , 0.6X + 08Y)
=cov(X,0.6X) + cov(X,0.8Y)
=0.6var(X,X)+0.8cov(X,Y)
=0.6var(X) + 0 (independence of X and Y)
=0.6 since X is standard normal.
now, var(X)=1, var(U) = 0.6^2 +0.8^2 = 1
so Corr(X,U) = cov(X,U)/sqrt(var(X)var(U)) = 0.6 / sqrt( 1*1) = 0.6
b) Let U = aX + bY ... pick arbitrary values for one of a or b, and run the calculations again and try to solve for the other given a particular correlation value.
corr(X,U) = cov(X,U)/sqrt(var(X)var(U))
cov(X,U)=cov(X , 0.6X + 08Y)
=cov(X,0.6X) + cov(X,0.8Y)
=0.6var(X,X)+0.8cov(X,Y)
=0.6var(X) + 0 (independence of X and Y)
=0.6 since X is standard normal.
now, var(X)=1, var(U) = 0.6^2 +0.8^2 = 1
so Corr(X,U) = cov(X,U)/sqrt(var(X)var(U)) = 0.6 / sqrt( 1*1) = 0.6
b) Let U = aX + bY ... pick arbitrary values for one of a or b, and run the calculations again and try to solve for the other given a particular correlation value.