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Let $\mu$ be a probability measure on $\Omega$, $X$ a random variable on $\Omega$. It is well known that the quantity $E[(X- c)^2]$is minimized over all $c\in \mathbb R$ by setting $c = E(X)$. What if we instead consider $E[|X-c|^p]$ for $p > 0$? If there a nice expression, in terms of $X$, for the value of $c$ which minimizes this expression?

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    For $p=1$ this is minimized by the median, but it may not be unique; in the limit as $p \rightarrow 0$ it is minimized by the mode, if the mode has positive probability.2012-11-09

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For example, consider the case $p =4$. You want to minimize $F(c) = E[(X-c)^4]$. So you take the derivative and set it to $0$: $ 0 = F'(c) = 4 E[(c-X)^3] = 4 (c^3 - 3 c^2 E[X] + 3 c E[X^2] - E[X^3])$ There is one real solution to this cubic (note that $F(c)$ is a strictly convex function of $c$ that goes to $+\infty$ as $c \to \pm \infty$). But a expression for it using radicals is not pleasant.

And for $p=6$ you'll get a quintic equation, which in general will not have a solution in radicals. For example, if $X$ is a binomial random variable with parameters $2$ and $1/3$, $F'(c) =6\,{c}^{5}-24+{\frac {200}{3}}\,c-80\,{c}^{2}+{\frac {160}{3}}\,{c}^{3 }-20\,{c}^{4}$ which has Galois group $S_5$.