I'm trying to get a good understanding of the higher moments of the Binomial Distribution; it's an important building block for more complex distributions so I want to get a strong intuition for this.
The thing that confused me was that lower p has higher kurtosis.
Looking at PMFs, it is clear the wing distributions are more "peaked" but the values don't seem to have a "fatter tail". If the kurtosis is the fourth power of the differences vs the mean, I would naively think the Binomial Dist with p=.5 has higher kurtosis; it seems more spread out!
I would have thought that the more extreme values of p are, the tighter values would be clustered around that. Because of this, the plot of variance makes sense to me.
So it makes sense to me that p=.5 has highest variance and least skew, but why would it also have the least kurtosis/thinnest tails?