User:Grick/Sandbox

Consider the mood (agreeableness, happiness, contentedness), insofar as such affects interpersonal (i.e., friends)  Suppose for a moment that an individual $$i$$'s "agreeableness" at time $$t$$ could be reasonably accurately* captured by a real number – call it $$e_i(t)$$. Assume boundedness of $$e_i(t)$$'s range (that is, there are minimum and maximum level agreeableness; i.e., there exists, at least on a theoretical level, a mood so foul that no other conceivable emotional state could be any less agreeable**), and normalize $$e_i(t)$$ to, say, the interval $$[0, 10]$$.  Now we can compare, across individuals, at least two aspects of the distribution of each individual's $$e_i(t)$$ over time:   $$\mu_{e}^{*}$$  $$\sigma_{e}^{*}$$ 
 * $$\mu_{e}^{}$$ – the mean (average) of agreeability, and
 * $$\sigma_{e}^{}$$ – the standard deviation (one measure of "spread" / variability / unpredictability)

yet it seems immensely plausible that $$(\mu_{e}^{*}, \sigma_{e}^{*}) = \underset{\mu_{e}^{}, \sigma_{e}^{}}{\operatorname{argmax}} \, f(\mu_{e}^{}, \sigma_{e}^{})$$

has $$\sigma_{e}^{*} > 0 $$ because dependability and predictability/spontaneity (imperturbable?)

In fact, more rigorously (and assuming continuity of $$f(\cdot, \cdot)$$), we might postulate that $$\exists \underline{\sigma_{e}^{}} : \sigma_{e}^{} \in [0, \underline{\sigma_{e}^{}}] \Rightarrow \frac{\partial {f(\cdot, \cdot)}}{\partial {\sigma_{e}^{}}} < 0 $$


 * – Clearly human emotions are not one-dimensional, so reducing any emotional variable to a single number necessarily entails significant loss of information.)
 * – Perhaps a rigorous argument for boundedness would invoke the finiteness of the human brain, limitations on the chemicals/receptors therein, etc.