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Consider an ordered pair of random variables $(x_i,y_i)$, and the following relationship:

$$\frac{\left(\sum_i^n x_i\right)^2}{\sum_i^n x_i y_i}\leq \sum_i^n\frac{x_i}{y_i}$$

Is there some useful way of restating the relationship in terms of statistical functions? Can the relationship between the two variables be summarized in some intuitive, informal way? The terms on LHS appear in deriving the least squares fit equation, but I don't recognize the RHS term.

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1 Answer 1

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It is not true in general. For instance, take $(x_1,x_2) = (-1,-1)$ and $(y_1,y_2) = (-1,2)$.

However, suppose all $x_i$ are positive and $\sum_i^n x_i y_i > 0$. If we rearrange the inequality a bit, we get: $$\frac{\sum_i^n x_i}{\sum_i^n\frac{x_i}{y_i}}\leq \frac{\sum_i^n x_i y_i}{\sum_i^n x_i} \,.$$ The left-hand side is a weighted harmonic mean with weights $x_i$. The right-hand side is a weighted arithmetic mean. Therefore, for positive $x_i$ the inequality is indeed always true.

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