How to measure the “Skewness” of a Probability Distribution? I looked at the "Fisher Measure" formula for "skewness of a probability distribution" (https://en.wikipedia.org/wiki/Skewness) - this is related to the expectation of the third moment for some transformed probability distribution.
How does this seemingly arbitrary formula using the expectation of the third moment specifically measure how skewed the probability distribution is? In other words, what is the motivation behind it?
Why does this not incorporate, for example, the second or fourth moment? Why even involve moments in this calculation to characterize the skewness of a distribution?
 A: I think the confusion here is that you are assuming that there is one universal way that statisticians define skewness. In actuality there are multiple measures of skewness - many of which are applicable depending on the exact context.
Some examples include:

Fisher's Moment Coefficient of Skewness: $\overline{\mu}_3 := \mathbb{E}\big{(}(\frac{X- \mu}{\sigma})^3\big{)}$
Mode Skewness: $\frac{1}{\sigma ^2}$(mean - mode)
Median Skewness: $\frac{3}{\sigma ^2}$(mean - median)

And there are many others. Therefore, your confusion as to why the skewness necessarily relates to the 3rd moment is resolved by the fact that this isn't the case. Both the mode and median skewness involve the mean (which is the first moment) and the standard deviation (which is a transformation of the second moment).
The key takeaway is that there are many different measures of skewness involving lots of different moments depending on which definition you are using.
The choice of definition ultimately comes down to the context and some subjective choice on which definition to go with. None of these definitions are "better" than any of the others in all situations. It's up to you to make a personal judgement.
Specifically looking at the Fisher's Moment Coefficient of Skewness (as this is the one that you seem to be referring to in your question), the reason that this works as one potential measure of skewness is that X- $\mu$ will tell you how far away the observations are from the mean and the fact that it is cubed means that you are able to distinguish whether or not $X$ is skewed to the left or the right of the mean. In other words, if the power was even, then you wouldn't be able to distinguish between whether or not $(X - \mu)^{2n}$ was positive or negative. Whereas $(X- \mu)^3$ is sufficient for us to know which direction the data has been skewed in).
A: If a probability distribution is symmetric around its mean ($\forall \delta: f(\mu - \delta) = f(\mu + \delta)$), then I assume you would agree that it should have a “skewness” of zero.  Odd moments have this property; even moments do not.
But the first standardized moment $\frac{E(x) - \mu}{\sigma}$ is always zero, which isn't useful.
That leaves the third standardized moment as the lowest-degree one that qualifies as a sensible measure of skewness: Zero for symmetric distributions, and non-zero for some distributions.
You could try using the fifth or seventh moment, or some other statistic that's not based on moments at all.  Some of them are listed on Wikipedia under “Other measures of skewness”.  It's just that Fisher has been traditionally established as the skewness coefficient, just as Pearson has been established as the measure of correlation between two variables.
