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Suppose I have the two datasets $x_i$ and $y_i$ with weights $w_{x,i}$ and $w_{y,i}$, respectively ($1 \le i \le N$). The weighted means are for example:

$$\bar x = \frac{\sum_i x_i w_{x,i}}{\sum_i w_{x,i}} \qquad \qquad \bar y = \frac{\sum_i y_i w_{y,i}}{\sum_i w_{y,i}}$$

What is the weight of the variable $z_i = x_i/y_i$? In particular, how can one compute $\bar z$?

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