The harmonic mean is one of several kinds of averages. Chapter 2 discussed how to compute the arithmetic mean, which is what most people typically think of when they compute an average. The harmonic mean, $H$, of the positive real numbers, $x_1,x_2,\dots ,x_n$, is defined as

The $F$ measure is the harmonic mean of precision and recall. Use this fact to derive Eq. (6.28) for $F$. In addition, write $F_{\beta }$ as a function of true positives, false negatives, and false positives.

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