* Gaussian

   \begin{displaymath} 	G(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]   \end{displaymath}

LATEXMODEPLUGIN_DEBUG = 0

  • Set LATEXFONTSIZE = footnotesize
blabla

\[ 	G(x; \beta\beta, \sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \]

   \begin{displaymath} 	G(x; \mu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]   \end{displaymath}

   \begin{displaymath} 	G(x; \nu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]   \end{displaymath}

   \begin{displaymath} A text first 5 inside latex\\     \mu_1xxxx = 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)  \end{displaymath}

  \begin{equation*}  \begin{array}{lr}    \mu_1x = 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{displaymath} \mathbf{X} = \left( \begin{array}{ccc} x_1 & x_2 & \ldots \\ x_3 & x_4 & \ldots \\ \vdots & \vdots & \ddots \end{array} \right) \end{displaymath}

 \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 &amp; \text{if } \quad x < 0\\ 0 &amp; \text{if } \quad x = 0\\ x &amp; \text{if } \quad x  > 0 \end{array} \right. \end{equation*}

- AlexanderFedotov - 04-Dec-2009

Sample Statistics

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] :

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

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

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
Latex rendering error!! dvi file was not created.

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