# Difference: AVFedotovLogA002 (1 vs. 7)

#### Revision 72010-02-01 - AlexanderFedotov

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The variances of the first few k-statistics are given by
\begin{array}{lr}
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var(k_1) = \frac{\kappa_1}{N} & (2.5.6a) \\ \ var(k_2) = \frac{\kappa_4}{N} + \frac{2\kappa_2^2}{N-1} & (2.5.6b) \\ \ var(k_3) = \frac{\kappa_6}{N} +\frac{9(\kappa_2\kappa_4 + \kappa_3^2) }{N-1} + \frac{6N\kappa_2^3}{(N-1)(N-2)} & \qquad (2.5.6c) \
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\mathrm{var}(k_1) = \frac{\kappa_1}{N} & (2.5.6a) \\ \ \mathrm{var}(k_2) = \frac{\kappa_4}{N} + \frac{2\kappa_2^2}{N-1} & (2.5.6b) \\ \ \mathrm{var}(k_3) = \frac{\kappa_6}{N} +\frac{9(\kappa_2\kappa_4 + \kappa_3^2) }{N-1} + \frac{6N\kappa_2^3}{(N-1)(N-2)} & \qquad (2.5.6c) \
\ldots \ \end{array}
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An unbiased estimator for is given by
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An unbiased estimator for is given by
In the special case of a normal parent population,
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an unbiased estimator for is given by
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an unbiased estimator for is given by

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Removing the averaging brackets around and

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on the right hand side of the equations, one gets the estimators on the left hand side:
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at the right hand side of the equations, one gets the estimators at the left hand side:

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### 2.8 Unbiased estimator for

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Replacing and in the expression (2.6.9) for by their estimators (2.7.4-5), gives the estimator for :

As , eq.(2.6.6), the same result (2.8.1) can be obtained using formula (2.5.7) by plugging into it the expressions (2.5.4b,c) for .

## References

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[1] Weisstein, Eric W. "Raw Moment." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RawMoment.html
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[1] Weisstein, Eric W. "Raw Moment." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RawMoment.html , pdf

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[2] Weisstein, Eric W. "Central Moment." http://mathworld.wolfram.com/CentralMoment.html
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[2] Weisstein, Eric W. "Central Moment." http://mathworld.wolfram.com/CentralMoment.html , pdf

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[3] Weisstein, Eric W. "Cumulant." http://mathworld.wolfram.com/Cumulant.html
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[3] Weisstein, Eric W. "Cumulant." http://mathworld.wolfram.com/Cumulant.html , pdf

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[4] Weisstein, Eric W. "Sample." http://mathworld.wolfram.com/Sample.html
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[4] Weisstein, Eric W. "Sample." http://mathworld.wolfram.com/Sample.html , pdf

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[5] Weisstein, Eric W. "Sample Raw Moment." http://mathworld.wolfram.com/SampleRawMoment.html
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[5] Weisstein, Eric W. "Sample Raw Moment." http://mathworld.wolfram.com/SampleRawMoment.html , pdf

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[6] Weisstein, Eric W. "Sample Mean." http://mathworld.wolfram.com/SampleMean.html
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[6] Weisstein, Eric W. "Sample Mean." http://mathworld.wolfram.com/SampleMean.html , pdf
[7] Weisstein, Eric W. "Sample Central Moment." http://mathworld.wolfram.com/SampleCentralMoment.html
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, pdf
[8] Weisstein, Eric W. "k-Statistic." http://mathworld.wolfram.com/k-Statistic.html
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, pdf
[9] Weisstein, Eric W. "Sample Variance." http://mathworld.wolfram.com/SampleVariance.html
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, pdf
[10] Weisstein, Eric W. "Standard Deviation." http://mathworld.wolfram.com/StandardDeviation.html
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[11] Weisstein, Eric W. "Sample Variance Distribution." http://mathworld.wolfram.com/SampleVarianceDistribution.html \ No newline at end of file
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#### Revision 62010-02-01 - AlexanderFedotov

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### 2.7 Unbiased estimators for and

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According to eq.(2.5.5a-c) the unbiased estimators for , and are given by , and (see eq.(2.5.4c)) , respectively. It is also important to have estimators for and .

Eqs. (2.4.3d) and (2.6.10) can be combined into a matrix equation

Solving this system of equations relative to and gives

Removing the averaging brackets around and on the right hand side of the equations, one gets the estimators on the left hand side:

## References

#### Revision 52010-02-01 - AlexanderFedotov

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According to eq.(2.4.3b), the expectation values for the variance and

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the statistic are (it agrees with the eq.(2.5.5b) too)
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the statistic are

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Surely, the eq.(2.6.7) agrees with the eq.(2.5.5b).

The variances are (eq.(4) in [11])

Eq.(23) in [11] gives a usefull expression for the :

## References

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[10] Weisstein, Eric W. "Standard Deviation." http://mathworld.wolfram.com/StandardDeviation.html
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[11] Weisstein, Eric W. "Sample Variance Distribution." http://mathworld.wolfram.com/SampleVarianceDistribution.html

#### Revision 42010-02-01 - AlexanderFedotov

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### 1.2 Central moments. Variance. Relations with the raw moments

A central moment is a moment taken about the mean [2] :
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\end{array}
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By definition the second central moment is the variance which is usually denoted as [10]:
The square root of the variance is called [10] the standard deviation :
The central moments are expressed via the raw moments using binomial transform [2] :
\begin{array}{lr} \displaystyle \mu_n = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} \mu'_k \, \mu'_1{}^{n-k} \quad .
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\end{array} In particular:
\begin{array}{lr}
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\mu_1 = 0 & (1.2.2a) \ \mu_2 = -\mu'_1{}^2 + \mu'_2 & (1.2.2b) \ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (1.2.2c) \ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 & \qquad (1.2.2d)
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\mu_1 = 0 & (1.2.4a) \ \mu_2 = -\mu'_1{}^2 + \mu'_2 & (1.2.4b) \ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (1.2.4c) \ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 & \qquad (1.2.4d)
\end{array} The raw moments are expressed via the central moments using
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(note and ):
\mu'_n = \sum_{k=0}^n \binom{n}{k} \mu_k \, \mu'_1{}^{n-k} \quad .
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\begin{array}{lr} \mu'_1 = \mu'_1 &\text{ (an identity)} \
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\mu'_2 = \mu_2{} + \mu'_1{}^2 & (1.2.3b) \ \mu'_3 = \mu_3 + 3 \mu_2 \mu'_1 + \mu'_1{}^3 & (1.2.3c) \
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\mu'_2 = \mu_2{} + \mu'_1{}^2 & (1.2.5b) \ \mu'_3 = \mu_3 + 3 \mu_2 \mu'_1 + \mu'_1{}^3 & (1.2.5c) \
\mu'_4 = \mu_4 + 4 \mu_3 \mu'_1 + 6 \mu_2 \mu'_1{}^2 + \mu'_1{}^4
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\end{array}
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\end{array}
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Since the is defined to be an unbiased estimator for the , one has , and then eqs.(1.3.3) give for the expectation values :
The variances of the first few k-statistics are given by
\begin{array}{lr}
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var(k_1) = \frac{\kappa_1}{N} & (2.5.5a) \\ \ var(k_2) = \frac{\kappa_4}{N} + \frac{2\kappa_2^2}{N-1} & (2.5.5b) \\ \ var(k_3) = \frac{\kappa_6}{N} +\frac{9(\kappa_2\kappa_4 + \kappa_3^2) }{N-1} + \frac{6N\kappa_2^3}{(N-1)(N-2)} & \qquad (2.5.5c) \
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var(k_1) = \frac{\kappa_1}{N} & (2.5.6a) \\ \ var(k_2) = \frac{\kappa_4}{N} + \frac{2\kappa_2^2}{N-1} & (2.5.6b) \\ \ var(k_3) = \frac{\kappa_6}{N} +\frac{9(\kappa_2\kappa_4 + \kappa_3^2) }{N-1} + \frac{6N\kappa_2^3}{(N-1)(N-2)} & \qquad (2.5.6c) \
\ldots \ \end{array}

An unbiased estimator for is given by

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In the special case of a normal parent population, an unbiased estimator for is given by
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### 2.6 Sample variance

The sample variance is the second central moment (2.4.1):

where is the sample mean (2.3.1) . It is commonly written as [9]
or sometimes

The square root of the sample variance is called the sample standard deviation [10]

Another widespread definition for the sample standard deviation is
where
By definition the statistic is an alternative name for the statistic defined above, see eq.(2.5.4b):

According to eq.(2.4.3b), the expectation values for the variance and the statistic are (it agrees with the eq.(2.5.5b) too)

## References

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[8] Weisstein, Eric W. "k-Statistic." http://mathworld.wolfram.com/k-Statistic.html
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[9] Weisstein, Eric W. "Sample Variance." http://mathworld.wolfram.com/SampleVariance.html

[10] Weisstein, Eric W. "Standard Deviation." http://mathworld.wolfram.com/StandardDeviation.html

#### Revision 32010-01-31 - AlexanderFedotov

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### 2.5 k-Statistics

The th k-statistic is [8] the unique symmetric unbiased estimator of the cumulant (see. e.g. eq.(1.3.4)), i.e., is defined so that

In addition, the variance
is a minimum compared to all other unbiased estimators of .

The k-statistics can be given in terms of the power sums (2.1.1)

Alternatively, they can be expressed via the sample mean (2.3.1) and central moments (2.4.1) by

The variances of the first few k-statistics are given by

An unbiased estimator for is given by

In the special case of a normal parent population, an unbiased estimator for is given by

## References

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[7] Weisstein, Eric W. "Sample Central Moment." http://mathworld.wolfram.com/SampleCentralMoment.html
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[8] Weisstein, Eric W. "k-Statistic." http://mathworld.wolfram.com/k-Statistic.html

#### Revision 22010-01-22 - AlexanderFedotov

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## 1 Population quantities

Let have a probability density function
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### 1.1 Moments = Raw Moments = Crude Moments. Mean.

A raw moment

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is a moment taken about 0 [1] :
\begin{array}{lr} \displaystyle
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\mu'_n = \langle x^n \rangle= \int x^n P(x) dx & \qquad (1)
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\mu'_n = \langle x^n \rangle= \int x^n P(x) dx & \qquad (1.1.1)
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By definition, the first moment is the mean of the distribution, ,

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### 1.2 Central moments and their relations with the raw moment

A central moment is a moment taken about the mean [2] :
\begin{array}{lr} \displaystyle \mu_n = \langle (x - \mu)^n \rangle= \int (x - \mu)^n P(x) \ dx \quad .
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\begin{array}{lr} \displaystyle \mu_n = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} \mu'_k \, \mu'_1{}^{n-k} \quad .
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\end{array} In particular:
\begin{array}{lr}
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\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 & \qquad (3d)
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\mu_1 = 0 & (1.2.2a) \ \mu_2 = -\mu'_1{}^2 + \mu'_2 & (1.2.2b) \ \mu_3 = 2 \mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (1.2.2c) \ \mu_4 = -3 \mu'_1{}^4 + 6 \mu'_1{}^2 \mu'_2 - 4 \mu'_1 \mu'_3 + \mu'_4 & \qquad (1.2.2d)
\end{array} The raw moments are expressed via the central moments using
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(note and ):
\mu'_n = \sum_{k=0}^n \binom{n}{k} \mu_k \, \mu'_1{}^{n-k} \quad .
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\begin{array}{lr} \mu'_1 = \mu'_1 &\text{ (an identity)} \
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\mu'_2 = \mu_2{} + \mu'_1{}^2 & (4b) \ \mu'_3 = \mu_3 + 3 \mu_2 \mu'_1 + \mu'_1{}^3 & (4c) \
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\mu'_2 = \mu_2{} + \mu'_1{}^2 & (1.2.3b) \ \mu'_3 = \mu_3 + 3 \mu_2 \mu'_1 + \mu'_1{}^3 & (1.2.3c) \
\mu'_4 = \mu_4 + 4 \mu_3 \mu'_1 + 6 \mu_2 \mu'_1{}^2 + \mu'_1{}^4
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\end{array}

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### 1.3 Cumulants and their expressions via moments

The characteristic function associated with the probability density function is defined as a Fourier transform [3] :
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The cumulants are then defined by
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They can be expressed through raw moments :
\begin{array}{lr}
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\kappa_1 = \mu'_1 & (7a) \ \kappa_2 = \mu'_2 - \mu'_1{}^2 & (7b) \ \kappa_3 = 2\mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (7c) \
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\kappa_1 = \mu'_1 & (1.3.3a) \ \kappa_2 = \mu'_2 - \mu'_1{}^2 & (1.3.3b) \ \kappa_3 = 2\mu'_1{}^3 - 3 \mu'_1 \mu'_2 + \mu'_3 & (1.3.3c) \
\kappa_4 = -6 \mu'_1{}^4 + 12 \mu'_1{}^2 \mu'_2 - 3 \mu'_2{}^2
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-4 \mu'_1 \mu'_3 + \mu'_4 & \qquad (7d) \
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-4 \mu'_1 \mu'_3 + \mu'_4 & \qquad (1.3.3d) \
\ldots & \ \end{array} or in terms of central moments :
\begin{array}{lr}
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\kappa_1 = \mu & (8a) \ \kappa_2 = \mu_2 & (8b) \ \kappa_3 = \mu_3 & (8c) \ \kappa_4 = \mu_4 - 3 \mu_2{}^2 & \qquad (8d) \
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\kappa_1 = \mu & (1.3.4a) \ \kappa_2 = \mu_2 & (1.3.4b) \ \kappa_3 = \mu_3 & (1.3.4c) \ \kappa_4 = \mu_4 - 3 \mu_2{}^2 & \qquad (1.3.4d) \
\ldots \ \end{array} where is the mean and is the variance.
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## 2 Sample quantities

A sample is a subset of a population [4]

where is the size of the sample.

### 2.1 Power sums

Power sum is the sum of th powers of the sample elements:

### 2.2 Sample raw moments

The th sample raw moment is defined as [5]

They are related to the power sums (2.1.1) by
They are unbiased estimators of the population raw moments (1.1.1) :

### 2.3 Sample mean

The sample mean [6] is defined by

It is equal to the sample first moment (2.2.1),
It is an unbiased estimator for the population mean (1.1.2)

### 2.4 Sample central moments

The th sample central moment is defined as [7]

where is the sample mean . The first few sample central moments are related to power sums by

In terms of the population central moments, the expectation values of the first few sample central moments are

## References

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[1] Wolfram MathWorld Raw Moment
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[1] Weisstein, Eric W. "Raw Moment." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RawMoment.html

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[2] Wolfram MathWorld Central Moment
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[2] Weisstein, Eric W. "Central Moment." http://mathworld.wolfram.com/CentralMoment.html

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[3] Wolfram MathWorld Cumulant
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[3] Weisstein, Eric W. "Cumulant." http://mathworld.wolfram.com/Cumulant.html

[4] Weisstein, Eric W. "Sample." http://mathworld.wolfram.com/Sample.html

[5] Weisstein, Eric W. "Sample Raw Moment." http://mathworld.wolfram.com/SampleRawMoment.html

[6] Weisstein, Eric W. "Sample Mean." http://mathworld.wolfram.com/SampleMean.html

[7] Weisstein, Eric W. "Sample Central Moment." http://mathworld.wolfram.com/SampleCentralMoment.html

#### Revision 12010-01-22 - AlexanderFedotov

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-- AlexanderFedotov - 04-Dec-2009

# Sample Statistics

## Population quantities

Let have a probability density function

### Moments = Raw Moments = Crude Moments

A raw moment (or just a moment, or a crude moment ) is a moment taken about 0 [1] :

### Central moments and their relations with the raw moment

A central moment is a moment taken about the mean [2] :

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

In particular:
The raw moments are expressed via the central moments using inverse binomial transform [1] (note and ):
In particular:

### Cumulants and their expressions via momemts

The characteristic function associated with the probability density function is defined as a Fourier transform [3] :

The cumulants are then defined by
They can be expressed through raw moments :
or in terms of central moments :
where is the mean and is the variance.

## References

[1] Wolfram MathWorld Raw Moment

[2] Wolfram MathWorld Central Moment

[3] Wolfram MathWorld Cumulant

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