Let $X$ be the random variable. $E(X)$ is the expected value of $X$
Then
$Var(X)$ = $E(X^2)$ − $[E(X)]^2$
where $Var(X)$ is the variance of $X$
Then how to represent skewness(X) in terms of the expected value.
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$\begingroup$ en.wikipedia.org/wiki/Skewness#Definition, replace $\mu$ with $\mathbb{E}(X)$. $\endgroup$– Richard HardyCommented Feb 20, 2020 at 13:17
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1$\begingroup$ You cannot, because skewness is a completely different property. Indeed, by adding a constant to $X$ you can make $E(X)$ anything you like without changing the skewness. $\endgroup$– whuber ♦Commented Feb 20, 2020 at 14:48
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1 Answer
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Every moment/standardised moment is expressed in terms of some kind of expected value. For example, you've expressed the variance using expected value. But you couldn't do it using only the expected value of $X$, i.e. $E[X]$, since you cannot express $E[X^2]$ using $E[X]$ only. Similarly, the skewness can be expressed in terms of expectations as follows: $$E\left[\left(\frac{(X-E[X])^3}{\operatorname{var}(X)^{3/2}}\right)\right]$$
But, it cannot be expressed using only $E[X]$.