If a random variable $X$ has a degenerate distribution, that is it takes a given value $k$ with probability $1$ and every other value with probability $0$, what is the skewness and kurtosis of $X$? Wikipedia states that these values are "undefined," but, intuitively, it would seem that the skewness should be $0$ (the distribution can't be skewed if it only takes one value). I'm not quite sure about Kurtosis.
Can somebody explain the reasoning behind these choices? Here's the wikipedia article I'm referring to: https://en.wikipedia.org/wiki/Degenerate_distribution
https://en.wikipedia.org/wiki/Skewness#Pearson's_moment_coefficient_of_skewness https://en.wikipedia.org/wiki/Kurtosis#Pearson_moments
If you look at the common definition of Skewness and Kurtosis, it is not only related to the centralized moments (which is $0$ for a degenerate distribution), it is about the standardized moment - they are standardized by some power of standard deviation / variance to make it unit-less. The variance of a degenerate distribution is $0$ also, so it is natural to set its skewness and kurtosis as undefined.
(There maybe other reasoning which people do not assign the value to be $0$. The above is just my first instinct.)