canCan someone provide an intuition on why the higher moments of a probability distribution p(x)$p_X$, like the third and fourth moments, correspond to skewness and kurtosis, respectively?
specifically Specifically, why does the deviation about the mean raised to the 3rdthird or 4thfourth power end up translating into a measure of skewness and kurtosis? Is Is there a way to relate this to the third or fourth derivatives of the function?
considerConsider this definition of skewness and kurtosis:
$Kurtosis(X) = E[(x - \mu_{X})^4] / \sigma^4$$$\begin{matrix} \text{Skewness}(X) = \mathbb{E}[(X - \mu_{X})^3] / \sigma^3, \\[6pt] \text{Kurtosis}(X) = \mathbb{E}[(X - \mu_{X})^4] / \sigma^4. \\[6pt] \end{matrix}$$
again,In these equations we raise the normalised value $(X-\mu)/\sigma$ to a power and take its expected value. It is not clear to me why raising $(x-\mu)^4$the normalised random variable to the power of four gives "peakedness" or why $(x-\mu)^3$raising the normalised random variable to the power of three should give skew"skewness". This seems magical and mysterious.
Edit: quick followup. what is the advantage of defining moments about the mean and not the median for metrics like kurtosis? what are the properties of estimators like:
$MedianKurtosis(X) = E[(x - \tilde{x})^4] / \sigma^4$
where $\tilde{x}$ is median. this would presumably be less sensitive to outliers in the distribution that throw off the mean and perhaps be a fairer measure of peaked-ness?!
