Epanechnikov-like bump . PDF

Function

 context $x_0,d:{\mathbb R}$ definition $k_n:{\mathbb N}_{\ge 0}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ definition $k_n(x):=\begin{cases} \dfrac{1}{2d}\left(1+\dfrac{1}{2n}\right)\left(1-\left(\dfrac{x-x_0}{d}\right)^{2n}\right) &\hspace{.5cm} \mathrm{if}\hspace{.5cm} \vert x \vert\le 1 \\\\ 0 \hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$

Discussion

$\lim_{n\to\infty}k_n$ is the normed $x_0$-centered rectangle of height $\frac{1}{2d}$.

We can of course linearly combine several such $k$-bumps and then normalize to obtain new PDF's.

Theorems

$\int_{x_0-d}^{x_0+d}\left(\dfrac{x}{d}\right)^{2m} k_n(x)\,{\mathrm d}x=\dfrac{1}{2(n+m)+1}\dfrac{2n+1}{2m+1}$

Code

P[n_, d_, x0_, x_] = 1/(2 d) (1 + 1/(2 n)) (1 - ((x - x0)/d)^(2 n));

Integrate[P[n, d, x0, x], {x, x0 - d, x0 + d}] // Expand

Manipulate[
Plot[
P[n, d, x0, x], {x, -3, 4}
, PlotRange -> {0, 1}, Filling -> Axis]
, {{n, 3}, 0, 50, 1}, {{d, 1}, 0, 3}, {{x0, 2}, -4, 4}]

Reference

Wikipedia: Epanechnikov-Kern