# Now when we take Z 1 </msub> = X + Y 1 </msub> and

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 ${\sigma }_{y}$?
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Step 1
By definition of correlation,
$\text{corr}\left({Z}_{1},{Z}_{2}\right)\equiv \frac{\text{Cov}\left({Z}_{1},{Z}_{2}\right)}{\sqrt{\text{Var}\left({Z}_{1}\right)\text{Var}\left({Z}_{2}\right)}}.$
The numerator is
$\text{Cov}\left({Z}_{1},{Z}_{2}\right)=\text{Cov}\left(X+{Y}_{1},X+{Y}_{2}\right)=\text{Var}\left(X\right)+\text{Cov}\left({Y}_{1},X\right)+\text{Cov}\left(X,{Y}_{2}\right)+\text{Cov}\left({Y}_{1},{Y}_{2}\right).$
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,
$\text{Cov}\left({Z}_{1},{Z}_{2}\right)=\text{Var}\left(X\right)\phantom{\rule{0ex}{0ex}}\text{Var}\left({Z}_{1}\right)=\text{Var}\left(X\right)+\text{Var}\left({Y}_{1}\right)\phantom{\rule{0ex}{0ex}}\text{Var}\left({Z}_{2}\right)=\text{Var}\left(X\right)+\text{Var}\left({Y}_{2}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\text{corr}\left({Z}_{1},{Z}_{2}\right)=\frac{{\sigma }_{x}^{2}}{{\sigma }_{x}^{2}+{\sigma }_{y}^{2}}.$