If I have that $X$ and $Y$ are independent r.v with mean 0 and variance $\sigma^2$ and I have that
$Z=X\sin\alpha + Y\cos\alpha \quad$, what is then the covariance and the correlation between X and Z?
I know that $cov(X,Z)=E[X(X\sin\alpha + Y\cos\alpha)]$
I dont know now how to proceed with covariance and correlation. Could somebody help me please?
Best Answer
You have :
$$cov(X,Z)=E[X(X\sin\alpha + Y\cos\alpha)]\\ =\sin\alpha\times E(X^2)+\cos\alpha\times E(XY)\\ =\sigma^2\sin\alpha+\cos\alpha\times cov(X,Y)\\ =\sigma^2\sin\alpha \text{ (because $X$ and $Y$ are independant)}$$
And you can also get the correlation : $$R=\sin\alpha $$