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Let $\mu$ be a probability measure, $f$ a function taking values in $[0,1]$. I am trying to determine the sign of the expression

$$3\left( \int f^2 d\mu \right)\left( \int f d\mu \right) - 2 \left( \int f d\mu \right)^3 -\left( \int f^3 d\mu \right).$$

Under what conditions is this either non-positive or non-negative?

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    Let $m_k = \int f^k \mathrm{d} \mu$ the the moment, then the expression at hand is $$ - \int \left( f - m_1 \right)^3 \mathrm{d} \mu $$ i.e., it is a negative of the third central moment of $X=f$. The expression is positive that $X$ is negatively skewed, zero when $X$ is symmetric, and negative when $X$ is positively skewed.2012-06-29

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