Central Moment
Definition and Overview
In probability theory and statistics, the central moment is a measure of the shape of a probability distribution. It is a specific type of moment, which is a quantitative measure of the shape of a function. Central moments are particularly useful because they provide information about the variability and symmetry of a distribution around its mean.
The central moment of order \( k \) for a random variable \( X \) with mean \( \mu \) is defined as: \[ \mu_k = \mathbb{E}[(X - \mu)^k] \] where \( \mathbb{E} \) denotes the expected value operator. The first central moment is always zero, and the second central moment is the variance of the distribution.
Mathematical Definition
For a random variable \( X \) with a probability density function \( f(x) \), the \( k \)-th central moment is given by: \[ \mu_k = \int_{-\infty}^{\infty} (x - \mu)^k f(x) \, dx \] where \( \mu \) is the mean of \( X \), defined as: \[ \mu = \int_{-\infty}^{\infty} x f(x) \, dx \]
In the case of a discrete random variable, the \( k \)-th central moment is: \[ \mu_k = \sum_{i=1}^{n} (x_i - \mu)^k P(X = x_i) \] where \( P(X = x_i) \) is the probability that \( X \) takes the value \( x_i \).
Properties of Central Moments
Central moments have several important properties:
1. **Translation Invariance**: Central moments are invariant under translation of the random variable. If \( Y = X + c \) for some constant \( c \), then the central moments of \( Y \) are the same as those of \( X \).
2. **Symmetry**: For symmetric distributions, all odd central moments are zero. This is because the distribution is symmetric around its mean, causing the contributions from \( (X - \mu)^k \) for odd \( k \) to cancel out.
3. **Scaling**: If \( Y = aX \) for some constant \( a \), then the \( k \)-th central moment of \( Y \) is \( a^k \) times the \( k \)-th central moment of \( X \).
Specific Central Moments
First Central Moment
The first central moment is always zero: \[ \mu_1 = \mathbb{E}[X - \mu] = \mathbb{E}[X] - \mu = 0 \]
Second Central Moment
The second central moment is the variance: \[ \mu_2 = \mathbb{E}[(X - \mu)^2] = \sigma^2 \] where \( \sigma^2 \) is the variance of \( X \). The square root of the variance is the standard deviation.
Third Central Moment
The third central moment is related to the skewness of the distribution: \[ \mu_3 = \mathbb{E}[(X - \mu)^3] \] Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values, while negative skewness indicates a distribution with an asymmetric tail extending towards more negative values.
Fourth Central Moment
The fourth central moment is related to the kurtosis of the distribution: \[ \mu_4 = \mathbb{E}[(X - \mu)^4] \] Kurtosis is a measure of the "tailedness" of the probability distribution. Higher kurtosis indicates more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly sized deviations.
Applications of Central Moments
Central moments are used in various fields such as:
1. **Statistical Inference**: Central moments are used to estimate parameters of probability distributions and to test hypotheses about the shape of distributions.
2. **Signal Processing**: In signal processing, central moments are used to characterize the shape and variability of signals.
3. **Econometrics**: Central moments are used in econometrics to model the distribution of economic variables and to assess the risk and return of financial assets.
4. **Physics**: In physics, central moments are used to describe the distribution of particle velocities in a gas and to analyze the shape of waveforms.
Higher-Order Central Moments
Higher-order central moments (beyond the fourth) are less commonly used but can provide additional information about the shape of a distribution. For example, the fifth central moment can provide insights into the asymmetry of the tails of the distribution, while the sixth central moment can provide information about the peakedness and tail behavior.
Calculation Methods
Central moments can be calculated using various methods, including:
1. **Analytical Methods**: For certain distributions, central moments can be calculated analytically using integral calculus.
2. **Numerical Methods**: For more complex distributions, central moments can be estimated using numerical integration or simulation techniques.
3. **Empirical Methods**: For sample data, central moments can be estimated using sample moments. The sample central moment of order \( k \) is given by: \[ \hat{\mu}_k = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^k \] where \( \bar{x} \) is the sample mean and \( n \) is the sample size.
Central Moments in Multivariate Distributions
For multivariate distributions, central moments can be extended to higher dimensions. The central moment of order \( (k_1, k_2, \ldots, k_d) \) for a \( d \)-dimensional random vector \( \mathbf{X} = (X_1, X_2, \ldots, X_d) \) with mean vector \( \boldsymbol{\mu} = (\mu_1, \mu_2, \ldots, \mu_d) \) is defined as: \[ \mu_{k_1, k_2, \ldots, k_d} = \mathbb{E}[(X_1 - \mu_1)^{k_1} (X_2 - \mu_2)^{k_2} \cdots (X_d - \mu_d)^{k_d}] \]
Examples of Central Moments
Normal Distribution
For a normal distribution \( N(\mu, \sigma^2) \), the central moments are well-known: - The first central moment is \( \mu_1 = 0 \). - The second central moment is \( \mu_2 = \sigma^2 \). - The third central moment is \( \mu_3 = 0 \) (since the normal distribution is symmetric). - The fourth central moment is \( \mu_4 = 3\sigma^4 \).
Exponential Distribution
For an exponential distribution with rate parameter \( \lambda \), the central moments are: - The first central moment is \( \mu_1 = 0 \). - The second central moment is \( \mu_2 = \frac{1}{\lambda^2} \). - The third central moment is \( \mu_3 = \frac{2}{\lambda^3} \). - The fourth central moment is \( \mu_4 = \frac{9}{\lambda^4} \).
See Also
