I have 2 random variables X and Y and I know the following about them:
E(X) = 50 and Var(X) = 16
Furthermore Y is defined as:
Y = 0,5 X + 20
I am looking for the correlation between X and Y.
Corr(X,Y) = Cov(X,Y) / [Var(X)^(1/2) * Var(Y)^(1/2)]
I already figured out that E(Y) = E(0,5X + 20) = 20 + 25 = 45
and I also know that the covariance is Cov(X,Y) = E(XY) - E(X)E(Y)
Now, here is the question: How can I get E(XY)? I know that you can calculate it when you have a contingency table. However, the excercise does not give any information about acutal values of X and Y. Moreover, I know that if the variables are independent you can calculate E(XY) by multiplying X and Y's means. But I think Y must depend upon X because it actually contains it.
Thanks for your help! :)
to make the solution easier to understand I will add the following rules that lead to this result:
Var(aX + b) = a^2 Var(X)
Cov(a1X + b1, a2Y + b^2) = a1 a2 Cov(X,Y)
Cov(X,X) = Var(X)