Gaussian kernel density worked example ($\hat{f}(0)=0.337949\dots$)

Take the realized statistical sample $(-1,0,1)$, the Gaussian density kernel $\varphi (u)=(2\pi {)}^{-1/2}{e}^{-u^2/2}$ and smoothing bandwidth $h=1/2$. ThenAt $x=0$, the three contributions are $0.03599398$, $0.26596152$, and $0.03599398$, because $\varphi (0)=0.39894228$ and $\varphi (2)=e^{-2}/\sqrt{2\pi }=0.05399097$. Their sum isThis is a probability density function value, not the probability of observing exactly zero. Each of the three component curves has area $1/3$, despite having a different height at zero.

Let $\Phi$ be the cumulative distribution function of the standard normal distribution. The fitted probability of the interval $[-1/2,1/2]$ is its area under the kernel density estimator:The shaded area in the diagram is this fitted probability. Its agreement to a few digits with $1/3$ is incidental: the empirical measure gives this interval mass $1/3$ exactly, whereas kernel density estimation redistributes that mass into neighboring intervals.