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! :)
Solution:
$$\mathsf{Corr(X,aX+b)}=\frac{\mathsf{Cov(X,aX+b)}}{\sqrt{\mathsf{Var}X}\sqrt{\mathsf{Var}(aX+b)}}=\frac{a\mathsf{Cov}(X,X)}{\sqrt{\mathsf{Var}X}\sqrt{a^{2}\mathsf{Var}X}}=\frac{a}{|a|}\in\{-1,1\}$$
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)
and
Cov(a1X + b1, a2Y + b^2) = a1 a2 Cov(X,Y)
finally,
Cov(X,X) = Var(X)
Shortcut (so not really an answer, but too much for a comment):
If $\mathsf{Var}(X)>0$ and $a\neq0$ then:
$$\mathsf{Corr(X,aX+b)}=\frac{\mathsf{Cov(X,aX+b)}}{\sqrt{\mathsf{Var}X}\sqrt{\mathsf{Var}(aX+b)}}=\frac{a\mathsf{Cov}(X,X)}{\sqrt{\mathsf{Var}X}\sqrt{a^{2}\mathsf{Var}X}}=\frac{a}{|a|}\in\{-1,1\}$$
In words: if $X$ is not degenerated with $\mathsf EX^2<\infty$ and $a\neq0$ then the correlation between $X$ and $Y=aX+b$ will be $1$ if $a>0$ and will be $-1$ if $a<0$.