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Numerically speaking

Empirical density functions

Definitions

The outcome of $N$ measurements of $x$ yields the sequence $x_{i}\,,\, i=1,\ldots,\, N$ (Sample measurements). Without any additional assumptions the probability density function (PDF) is \begin{equation} \rho_{e}(x)=\frac{1}{N}\sum_{i=1}^{N}\delta(x-x_{i})\label{eq:rhoe} \end{equation} Here $\delta(x)$ is the Dirac delta function [1]. The density (\ref{eq:rhoe}) is also called a raw density function [2]. The PDF (\ref{eq:rhoe}) is obviously normalized.

The corresponding cumulative probability function is

\begin{equation} P_{e}(x)=\frac{1}{N}\sum_{i=1}^{N}H(x-x_{i}) \end{equation} Here $H(x)$ is the Heaviside step function [3]. For any function $f(x)$ this PDF gives the following average values \begin{equation} \left\langle \, f\,\right\rangle =\intop_{a}^{b}f(x)\rho_{e}(x)dx=\frac{1}{N}\sum_{i=N}^{N}f(x_{i}) \end{equation} In particular, the moments are \[ \left\langle x^{n}\right\rangle =\frac{1}{N}\sum_{i=N}^{N}x_{i}^{n} \]

Remarks

 

References

  1. The Dirac delta function.
  2. Kernel bandwidth optimization in spike rate estimation.
  3. The Heaviside step function.

 

© Nikolai Shokhirev, 2012-2017

email: nikolai(dot)shokhirev(at)gmail(dot)com

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