Sandbox/SandboxArchive Web>AlexanderFedotovSandbox4 (2020-08-18, TWikiAdminUser)

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$A text first 5 inside latex\\ \mu_1 = 0 \qquad\qquad\qquad\qquad (3a) \\ \mu_2 = -\mu'_1{}^2 + \mu'_2 \qquad\qquad\qquad (3b) \\ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 \qquad\qquad (3c) \\ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 \qquad (3d)$

$\begin{equation*} \begin{array}{lr} \mu_1 = 0 & (3a) \\ \mu_2 = -\mu'_1{}^2 + \mu'_2 & (3b) \\ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (3c) \\ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 &(3d) \end{array} \end{equation*}$

$\begin{array}{lr} \mu_1 = 0 & (3a) \\ \mu_2 = -\mu'_1{}^2 + \mu'_2 & (3b) \\ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (3c) \\ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 &(3d) \end{array}$

$\begin{equation*} \mathbf{X} = \left( \begin{array}{ccc} x_1 & x_2 & \ldots \\ x_3 & x_4 & \ldots \\ \vdots & \vdots & \ddots \end{array} \right) \end{equation*}$

$\begin{equation*} \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22}) \end{array} \right) \cdot \left( \begin{array}{c} x\\y \end{array} \right) = \left( \begin{array}{c} C\\D \end{array} \right) \end{equation*}$

$\begin{equation*} \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \qquad \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \end{equation*}$

$\begin{equation*} \vert x \vert = \left\{ \begin{array}{rl} -x & \text{if } \quad x < 0\\ 0 & \text{if } \quad x = 0\\ x & \text{if } \quad x > 0 \end{array} \right. \end{equation*}$

- AlexanderFedotov - 04-Dec-2009

Sample Statistics

Sample Statistics

Sections:

Population quantities
References

Population quantities

Let $x$ have a probability density function $ P(x) $

Moments = Raw Moments = Crude Moments

A raw moment $\mu'_n$ (or just a moment, or a crude moment ) is a moment taken about 0 [1] :

$\mu'_n = \\ \langle x^n \rangle = \int x^n P(x) dx \quad . \qquad (1)$

Central moments

A central moment $\mu _n$ is a moment taken about the mean $\mu = \mu' _1$ [2] :

$\mu_n = \langle (x - \mu)^n \rangle= \int (x - \mu)^n P(x) dx \quad . \qquad (2)$

Translation $\mu_n \to \mu'_n$

The central moments are expressed via the raw moments using binomial transform [2] :

$\mu_n = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} \mu'_k \, \mu'_1{}^{n-k} \quad . \qquad(3)$

In particular:

$\mu_1 = 0 \qquad\qquad\qquad\qquad (3a) \\ \mu_2 = -\mu'_1{}^2 + \mu'_2 \qquad\qquad\qquad (3b) \\ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 \qquad\qquad (3c) \\ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 \qquad (3d)$

Translation $\mu'_n \to \mu_n$

The raw moments are expressed via the central moments using inverse binomial transform [1] (note $\mu_0 = 1$ and $\mu_1 = 0$ ):

$\mu'_n = \sum_{k=0}^n \binom{n}{k} \mu_k \, \mu'_1{}^{n-k} \quad . \qquad(4)$

In particular:

$\mu'_1 = \mu'_1 \qquad\qquad\qquad\qquad \text{ (an identity) } \\ \mu'_2 = \mu_2{} + \mu'_1{}^2 \qquad\qquad\qquad (4b) \\ \mu'_3 = \mu_3 + 3 \mu_2 \mu'_1 + \mu'_1{}^3 \qquad\qquad (4c) \\ \mu'_4 = \mu_4 + 4 \mu_3 \mu'_1 + 6 \mu_2 \mu'_1{}^2 + \mu'_1{}^4 \qquad (4d) \\$

Cumulants

The characteristic function $\phi(t)$ associated with the probability density function $ P(x) $ is defined as a Fourier transform [3] :

$\phi(t) = \int_{-\infty}^{\infty} e^{itx} P(x) dx \quad . \qquad(5)$

The cumulants $\kappa_n$ are then defined by

$\ln \phi(t) \equiv \sum_{n=1}^{\infty} \kappa_n \frac{(it)^n}{n!} \quad . \qquad(6)$

They can be expressed through raw moments $\mu'_n$ :

$\kappa_1 = \mu'_1 \qquad\qquad\qquad\qquad (7a) \\ \kappa_2 = \mu'_2 - \mu'_1{}^2 \qquad\qquad\qquad (7b) \\ \kappa_3 = 2\mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 \qquad\qquad (7c) \\ \kappa_4 = -6 \mu'_1{}^4 + 12 \mu'_1{}^2 \mu'_2 - 3 \mu'_2{}^2 -4 \mu'_1 \mu'_3 + \mu'_4 \qquad (7d) \\ \ldots \\$

or in terms of central moments $\mu_n$ :

$\kappa_1 = \mu \qquad\qquad (8a) \\ \kappa_2 = \mu_2 \qquad\qquad (8b) \\ \kappa_3 = \mu_3 \qquad\qquad (8c) \\ \kappa_4 = \mu_4 - 3 \mu_2{}^2 \qquad (8d) \\ \ldots \\$

where $\mu$ is the mean and $\sigma^2 \equiv \mu_2$ is the variance.

References

[1] Wolfram MathWorld Raw Moment

[2] Wolfram MathWorld Central Moment

[3] Wolfram MathWorld Cumulant
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Topic revision: r6 - 2020-08-18 - TWikiAdminUser

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