Statistical Moments

see also:

• statistics
• Random Variables

Statistical moments are quantitative measures that capture various aspects of the shape and characteristics of a probability distribution. They are fundamental in the fields of statistics and probability, providing insights into the distribution’s location, scale, skewness, and kurtosis, among other features. Each moment corresponds to a particular derivative of a generating function, such as the moment-generating function or the cumulant-generating function.

Here’s a detailed breakdown of the first few moments and their significance:

1. First Moment: Mean

The first moment about the origin provides the mean ($\mu$) of the distribution, which is a measure of central tendency. It is defined as: $\mu =E[X]={\int }_{-\infty }^{\infty }xf(x)dx$ for a continuous variable $X$ with density $f(x)$, or $\mu =E[X]={\sum }_i{x}_i{p}_i$ for a discrete variable, where $x_i$ are the values and $p_i$ their respective probabilities.

2. Second Moment: Variance

The second moment about the mean provides the variance (${\sigma }^2$), which measures the dispersion of the distribution around its mean. It is computed as: ${\sigma }^2=E[(X-\mu {)}^2]={\int }_{-\infty }^{\infty }(x-\mu {)}^{2}f(x)dx$ for continuous variables, or ${\sigma }^2=E[(X-\mu {)}^2]={\sum }_i(x_i-\mu {)}^2{p}_i$ for discrete variables.

3. Third Moment: Skewness

The third central moment is used to calculate the skewness of the distribution, which measures the asymmetry of the distribution around its mean. Skewness is normalized by the standard deviation to the third power, thus: $\text{Skewness}=\frac{E[(X-\mu {)}^3]}{{\sigma }^3}$ A positive skew indicates a distribution with an extended tail to the right, while a negative skew indicates a tail to the left.

4. Fourth Moment: Kurtosis

The fourth central moment is used to define the kurtosis, which measures the “tailedness” of the distribution. Kurtosis is often normalized by the square of the variance (standard deviation squared), giving: $\text{Kurtosis}=\frac{E[(X-\mu {)}^4]}{{\sigma }^4}$ A higher kurtosis than that of a normal distribution (which has a kurtosis of 3) indicates a distribution with heavier tails and a sharper peak, termed leptokurtic.

Higher Moments

Higher moments ($n^{th}$ moments) about the mean can be similarly defined and are used to describe additional aspects of the distribution’s shape: $E[(X-\mu {)}^n]$ They become progressively more complex and harder to interpret but can be crucial in certain statistical models, particularly in finance and signal processing.

Statistical moments are not only foundational in descriptive statistics but also serve as the building blocks for various forms of statistical inference, helping to characterize distributions for hypothesis testing, model fitting, and more. For deeper statistical analysis, exploring the relationships between moments and other distribution characteristics or leveraging them in the form of Cumulants and moment generating functions can provide further insights.

statistical moments and m-point correlators

Statistical moments and (m)-point correlators are fundamental concepts in statistics and physics, especially in the study of quantum field theory (QFT) and statistical mechanics. They provide critical insights into the distribution and correlations of physical quantities in various systems. Understanding these concepts is essential for analyzing the behavior of quantum fields and many-body systems, as well as for characterizing probability distributions in a statistical context.

Statistical Moments

In statistics, moments are quantitative measures related to the shape of a distribution. The (n)-th moment of a probability distribution about a point (a) is defined as the expected value of ((X-a)^n), where (X) is a random variable. Moments about the mean ((a = \mu), the mean of the distribution) are particularly informative:

• The first moment about the origin ((a = 0)) gives the mean of the distribution, providing a measure of its central tendency.
• The second central moment ((a = \mu)) is the variance, measuring the distribution’s spread.
• The third central moment (normalized by the variance to the power of (3/2)) gives the skewness, indicating the distribution’s asymmetry.
• The fourth central moment (normalized by the variance squared) gives the kurtosis, reflecting the “tailedness” of the distribution or the propensity of the distribution to produce outliers.

(m)-Point Correlators

In the context of QFT and statistical mechanics, (m)-point correlators (also known as Green’s functions or correlation functions) are used to describe the correlations between field values at (m) different points in space-time. They are fundamental to understanding the interactions and dynamics within a field theory.

An (m)-point correlator is defined as the expectation value (or vacuum expectation value in QFT) of the product of field operators evaluated at (m) different points:

$$G(x_1,x_2,...,x_m)=⟨0|\hat{\varphi }(x_1)\hat{\varphi }(x_2)...\hat{\varphi }(x_m)|0⟩$$

where (\hat{\phi}(x_i)) is the field operator at point (x_i) and (| 0 \rangle) denotes the vacuum state.

• In QFT, these correlators describe the probability amplitudes for particles to propagate from one point to another and are instrumental in calculating scattering amplitudes and cross-sections using Feynman diagrams.
• In statistical mechanics, correlation functions measure how system properties at different points in space (or time) are related, providing insights into phase transitions, critical phenomena, and other collective behaviors.

Connection and Differences

• Nature: Statistical moments are primarily used in the analysis of probability distributions and are scalar quantities. (m)-Point correlators are used in field theories to describe quantum or statistical correlations between different points in space-time and are functional in nature.
• Application: Moments are fundamental to characterizing distributions in a wide range of contexts, from simple random processes to complex systems. (m)-Point correlators are specific to the study of dynamical systems described by field theories, where they capture the essence of interactions and correlations.
• Computation: The computation of moments in statistics often involves integration over probability distributions, while the calculation of (m)-point correlators in QFT might require perturbative expansions, Wick’s theorem, and the evaluation of Feynman diagrams.

Both statistical moments and (m)-point correlators serve as indispensable tools in their respective fields, enabling the quantitative analysis of complex systems and phenomena.