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rnorouzian
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Variance of difference of two correlated variables when working with random samples of each

I know that the variance of the difference of two correlated variables, $Y_1$ and $Y_2$ , is what the below formula shows, requiring $r$ the correlation between the two variables.

But suppose I have an amply large random sample of each of above variables, $Y_1^*$ and $Y_2^*$, in isolation.

Does variance of $Y_2^* - Y_1^*$ give me a good estimate of variance of the difference between $Y_1$ and $Y_2$?

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rnorouzian
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