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Reading Blitzstein and Hwang’s introduction to probability http://probabilitybook.net (really good so far!).

On page 172, they discuss why $E[|X-E[X]|$ isn’t used as the definition of variance. They mention that a primary reason is that the absolute value function isn’t differentiable at zero whereas the squaring function is differentiable everywhere.

In my adventures so far, I’m yet to encounter an instance where one needs to be able to differentiate the variance of X.

When is it useful to be able to differentiate Var(X) and what does it mean to differentiate the variance?

My guess-from-the-hip is that one might want to minimise/maximise variance?

Thanks in advance :)

apprentice9
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    Please refer to any of our (hundreds) of posts that differentiate the objective function in ordinary least squares regression in order to derive the solution. – whuber Aug 18 '21 at 15:16
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    This comment is about the function inside the expectation bracket, the integrand, and not the quantity outside which is possibly differentiable with respect to the decision procedure (e.g., an estimator) or the parameter of the distribution. For instance, the mean absolute deviation for a Normal $\mathcal N(\mu,\sigma^2)$ is $\sqrt{2/\pi}\sigma$, which is differentiable in $\sigma$ and $\mu$. – Xi'an Aug 18 '21 at 15:16
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    BTW, this point about lack of differentiability is (as your question hints) a bit of a red herring, because it's not terribly important. A far deeper explanation lurks in the Central Limit Theorem. This fundamental result identifies the variance as the *unique* (up to asymptotic equivalence) way to express *variability* when you are studying the averages of large numbers of independent values. For an indication of why that is, see https://stats.stackexchange.com/a/3904/919. – whuber Aug 19 '21 at 18:54

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