Moment (mathematics)

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Moment (mathematics) is a concept in mathematics that is used in various fields such as probability theory, statistics, physics, and engineering to provide a quantitative measure of the shape of a set of points. The moments of a function or a distribution can offer insight into its characteristics, such as its central tendency, variability, skewness, and kurtosis. Moments are, in essence, the weighted averages of a function's values raised to a power, with the weights being the function's values at specific points.

Definition

In a general sense, the nth moment of a real-valued continuous function f(x) of a real variable about a value c is given by the integral

M_n = \int (x - c)^n f(x) dx,

where n is a non-negative integer and the integral is taken over the domain of f. When c is the mean of the distribution, the moments are called central moments. The first central moment is always 0, as it is the mean of the distribution itself. The second central moment is known as the variance of the distribution, a measure of its spread.

In the context of probability distributions, the nth moment about zero (also called the raw moment) of a random variable X is the expected value of X^n, and is denoted as E[X^n].

Types of Moments

There are several types of moments used in different contexts:

• Raw moments: The moments about zero, which provide information about the distribution's shape relative to the origin.
• Central moments: The moments about the mean, which are used to describe the distribution's variability and shape without being influenced by its location.
• Standardized moments: These are derived from central moments but are normalized, making them dimensionless and allowing for comparison between different distributions. The third standardized moment is related to skewness, and the fourth to kurtosis.

Applications

Moments have a wide range of applications across various disciplines:

• In statistics, moments are used to describe the characteristics of distributions, such as the mean, variance, skewness, and kurtosis.
• In physics, moments describe the distribution of physical quantities in space, such as the moment of inertia, which characterizes an object's rotational inertia.
• In engineering, moments are applied in the analysis of beams to determine bending moments, which help in designing structures to withstand loads.

Calculating Moments

The calculation of moments can vary depending on the context and the specific function or distribution. For discrete distributions, moments are calculated as the sum of the products of the values and their probabilities raised to the nth power. For continuous distributions, moments are calculated using integrals.

See Also

• Probability distribution
• Expected value
• Variance
• Skewness
• Kurtosis

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