Suppose you have data $X$ from a normal distribution with $\mu_x$ and a standard deviation $\sigma_x$. Additionally you have data $Y_1$ and $Y_2$ both sampled from a normal distribution with $\mu_y$ and standard deviation $\sigma_y$. As $Y_1$ and $Y_2$ are not the same datasets the correlation is zero.
Now when we take $Z_1=X+Y_1$ and $Z_2=X+Y_2$, what can we say about the correlation coefficient between $Z_1$ and $Z_2$?
For this case, is it possible to find the correlation coefficient as function of $\sigma_x$ and $\sigma_y$?
If necessary we can assume $\mu_x=0$ and $\mu_y=0$.
Best Answer
By definition of correlation, $$\text{corr}(Z_1,Z_2)\equiv \frac{\text{Cov}(Z_1,Z_2)}{\sqrt{\text{Var}(Z_1)\text{Var}(Z_2)}}.$$
The numerator is
$$\text{Cov}(Z_1,Z_2)=\text{Cov}(X+Y_1,X+Y_2)=\text{Var}(X)+\text{Cov}(Y_1,X)+\text{Cov}(X,Y_2)+\text{Cov}(Y_1,Y_2).$$
We need information on how $X$ is correlated with $Y_1,Y_2$ to proceed further, but assuming $X,Y_1,Y_2$ are all uncorrelated,
$$\text{Cov}(Z_1,Z_2)=\text{Var}(X)\\ \text{Var}(Z_1)=\text{Var}(X)+\text{Var}(Y_1)\\ \text{Var}(Z_2)=\text{Var}(X)+\text{Var}(Y_2)\\ \implies \text{corr}(Z_1,Z_2)=\frac{\sigma^2_x}{\sigma^2_x+\sigma^2_y}. $$