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Epanechnikov-like bump . PDF
Set
context | $x_0,d:{\mathbb R}$ |
definition | $k:{\mathbb N}\to{\mathbb R}\to{\mathbb R}_{\ge 0}$ |
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} \left|x\right|\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 and normalize to gain PDF’s with several such $k$-bumps 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