If $X$ and $Y$ are random variables such that

$$\sigma^2_x = \sigma^2_{x, \text{unique}} + \sigma^{2}_{x,y,\text{shared}}$$

$$\sigma^2_y = \sigma^2_{y, \text{unique}} + \sigma^{2}_{x,y,\text{shared}}$$

Then what is: $$\operatorname{Var}(X-Y)$$

In a paper I have read recently they write:

$$\operatorname{Var}(X-Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) - 2\operatorname{Cov}(X,Y)$$ $$= \sigma^2_{x, \text{unique}} + \sigma^2_{x,y,\text{shared}} + \sigma^2_{y, \text{unique}} + \sigma^2_{x,y,\text{shared}} - 2\sigma^2_{x,y,\text{shared}}$$ $$= \sigma^2_{x, \text{unique}} + \sigma^2_{y, \text{unique}}$$

However, I am not fully convinced that $$\operatorname{Cov}(X,Y) = \sigma^2_{x,y,\text{shared}}$$

Is there an obvious proof of this I am missing?


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