Understanding "the kernel has zero mean"

Question

I am trying to understand kernel density estimation and found the graphic below illustrating different kernel functions on Wikipedia. I have no trouble reconciling it with the two statements

1. "the kernel integrates to one", i.e. $\int K(z)\mathrm{d}z=1$, and
2. "the kernel is symmetric around zero", i.e. $K(z)=K(-z)$.

But I've several times seen the statement "the kernel has zero mean" and I can't make sense out of it.

Does it mean

1. $\mathbb{E}[K(z)]=\int K(z)z\mathrm{d}z=0$ or
2. something else entirely?

Answer 1

The first statement is almost, but not entirely correct. $K$ is a density function and what we mean when we refer to the mean of the density function $K$ is $$\int K(z)z\mathrm{d}z.$$ Similarly, the variance of $K$ is $$\int K(z)z^2\mathrm{d}z.$$ So, it doesn't make any sense to write $\mathbb{E}[K(z)]$. Rather, the statement "the kernel has zero mean" should be formally written as $$\int K(z)z\mathrm{d}z=0.$$ (Thanks, Glen_b!)