Draft:Epanechnikov distribution

In probability theory and statistics, the Epanechnikov distribution, also known as the Epanechnikov kernel, is a continuous probability distribution that is defined on a finite interval. It is named after V. A. Epanechnikov, who introduced it in 1969 in the context of kernel density estimation.

Definition
A random variable has an Epanechnikov distribution if its probability density function is given by:


 * $$f(x) = \frac{3}{4c}\max\left(0, 1-\left(\frac{x}{c}\right)^2\right)$$

where $$c > 0$$ is a scale parameter. Setting $$c= \sqrt{5}$$ gives the unit variance probability distribution originally considered by Epanechnikov.

Cumulative distribution function
The cumulative distribution function (CDF) of the Epanechnikov distribution is:


 * $$F(x) = \frac{1}{2} + \frac{3x}{4c} - \frac{x^3}{4c^3}$$ for $$-c \leq x \leq c$$

Moments and other properties

 * Mean: $$E[X] = 0$$
 * Median: $$\text{Median}[X] = 0$$
 * Mode: $$\text{Mode}[X] = 0$$
 * Variance: $$\text{Var}[X] = \frac{c^2}{5}$$

Applications
The Epanechnikov distribution has applications in various fields, including:


 * Kernel density estimation: It is widely used as a kernel function in non-parametric statistics, particularly in kernel density estimation. In this context, it is often referred to as the Epanechnikov kernel. For more information, see Kernel functions in common use.

Related distributions

 * The Epanechnikov distribution can be viewed as a special case of a Beta distribution that has been shifted and scaled along the x-axis.