Glossary

8.3 Higher-order moments and central moments

3 min read•july 19, 2024

Moments in probability theory go beyond and variance, giving us a deeper understanding of random variables. Higher-order moments help describe distribution shapes, while central moments measure spread around the mean. These concepts are crucial for analyzing data and making predictions.

and are key measures derived from moments. Skewness tells us about asymmetry, while kurtosis reveals how peaked or flat a distribution is. Understanding these properties helps us compare different probability distributions and interpret real-world data more effectively.

Moments and Their Applications

Higher-order and central moments

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  • Higher-order moments generalize the concept of moments beyond mean and variance
    • nn-th raw moment of random variable XX defined as E[Xn]E[X^n]
    • nn-th central moment of random variable XX defined as E[(X−μ)n]E[(X - \mu)^n], where μ=E[X]\mu = E[X] is the mean
  • Calculate higher-order and central moments:
    • For discrete random variable, use summation formula: E[Xn]=∑xxn⋅P(X=x)E[X^n] = \sum_{x} x^n \cdot P(X = x) and E[(X−μ)n]=∑x(x−μ)n⋅P(X=x)E[(X - \mu)^n] = \sum_{x} (x - \mu)^n \cdot P(X = x)
    • For continuous random variable, use integration formula: E[Xn]=∫−∞∞xn⋅f(x)dxE[X^n] = \int_{-\infty}^{\infty} x^n \cdot f(x) dx and E[(X−μ)n]=∫−∞∞(x−μ)n⋅f(x)dxE[(X - \mu)^n] = \int_{-\infty}^{\infty} (x - \mu)^n \cdot f(x) dx

Skewness and kurtosis interpretation

  • Skewness measures asymmetry of probability distribution
    • Positive skewness indicates longer or fatter tail on right side of distribution (income distribution)
    • Negative skewness indicates longer or fatter tail on left side of distribution (stock returns during market crash)
    • Zero skewness indicates symmetric distribution (standard )
  • Kurtosis measures tailedness and peakedness of probability distribution
    • Higher kurtosis indicates heavier tails and more peaked distribution compared to normal distribution (financial returns)
    • Lower kurtosis indicates lighter tails and flatter distribution compared to normal distribution (uniform distribution)

Characterization of probability distributions

  • Third standardized moment, skewness coefficient, quantifies skewness of distribution
    • Skewness coefficient = E[(X−μ)3]/σ3E[(X - \mu)^3] / \sigma^3, where σ\sigma is standard deviation
  • Fourth standardized moment, kurtosis coefficient, quantifies kurtosis of distribution
    • Kurtosis coefficient = E[(X−μ)4]/σ4E[(X - \mu)^4] / \sigma^4
  • Comparing skewness and kurtosis coefficients of different probability distributions helps understand their relative shapes and tail behaviors
    • Exponential distribution has positive skewness and higher kurtosis than normal distribution
    • Uniform distribution has zero skewness and lower kurtosis than normal distribution

Relationship Between Moments

Raw vs central moments relationship

  • Relationship between raw moments and central moments derived using binomial expansion of (X−μ)n(X - \mu)^n
  • For first four central moments:
    1. E[(X−μ)1]=0E[(X - \mu)^1] = 0
    2. E[(X−μ)2]=E[X2]−μ2E[(X - \mu)^2] = E[X^2] - \mu^2
    3. E[(X−μ)3]=E[X3]−3μE[X2]+2μ3E[(X - \mu)^3] = E[X^3] - 3\mu E[X^2] + 2\mu^3
    4. E[(X−μ)4]=E[X4]−4μE[X3]+6μ2E[X2]−3μ4E[(X - \mu)^4] = E[X^4] - 4\mu E[X^3] + 6\mu^2 E[X^2] - 3\mu^4
  • In general, central moments can be expressed as function of raw moments and mean μ\mu
    • Variance (2nd central moment) expressed in terms of 2nd raw moment and squared mean
    • Skewness (3rd standardized moment) expressed in terms of first three raw moments and mean
    • Kurtosis (4th standardized moment) expressed in terms of first four raw moments and mean

Key Terms to Review (15)

Additivity of Moments: Additivity of moments refers to the principle that the moments of a random variable can be calculated by summing the moments of its independent components. This principle is essential for understanding how different random variables contribute to the overall behavior of a system, particularly in terms of their mean and higher-order moments, such as variance and skewness. It plays a significant role in assessing the characteristics of distributions and allows for simplification in calculations involving multiple random variables.
Binomial Distribution: The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. This distribution is essential for understanding random events that have two possible outcomes, like flipping a coin or passing a test, and it connects closely with the foundational concepts of probability, randomness, and statistical measures.
E(x^n): e(x^n) refers to the expected value of the nth power of a random variable X, representing a higher-order moment in probability theory. This term is crucial for understanding the behavior of random variables, as it helps in analyzing their distributions and their relationships with different statistical measures. It serves as a foundational concept for exploring higher-order moments and central moments, allowing insights into variance, skewness, and kurtosis in probability distributions.
E[(x - μ)^n]: The expression e[(x - μ)^n] represents the expected value of the n-th power of the deviation of a random variable x from its mean μ. This concept is central to understanding higher-order moments and central moments, as it helps in measuring how much the values of a random variable deviate from the mean and the degree of variability within a distribution.
Empirical Moments: Empirical moments are statistical measures that summarize the shape and characteristics of a dataset by calculating the mean, variance, skewness, and kurtosis based on observed data. These moments help in understanding how data is distributed and in identifying patterns, variability, and the presence of outliers within the dataset. They are essential for deriving insights into data behavior and provide a foundation for further statistical analysis.
Fourth Moment: The fourth moment is a statistical measure that quantifies the shape of a probability distribution, specifically relating to its tails and how spread out the values are around the mean. It is calculated as the average of the fourth powers of the deviations from the mean, providing insight into the distribution's kurtosis. High fourth moment values indicate heavy tails, suggesting a greater likelihood of extreme outcomes compared to a normal distribution.
Kurtosis: Kurtosis is a statistical measure that describes the shape of a probability distribution's tails in relation to its overall shape. It specifically provides insight into the extent of outliers in the data, highlighting whether the distribution is peaked (leptokurtic), flat (platykurtic), or normal (mesokurtic). Understanding kurtosis helps to analyze the behavior of data beyond just its mean and variance, shedding light on potential extremes and risks involved in the dataset.
Mean: The mean, often referred to as the average, is a measure of central tendency that quantifies the expected value of a random variable. It represents the balancing point of a probability distribution, providing insight into the typical outcome one can expect from a set of data or a probability distribution. The concept of the mean is essential in understanding various statistical properties and distributions, as it lays the foundation for further analysis and interpretation.
Method of Moments: The method of moments is a statistical technique used to estimate parameters of a probability distribution by equating sample moments to theoretical moments. This approach connects the properties of a distribution, such as mean and variance, with the observed data, allowing for parameter estimation in a straightforward manner. It is particularly useful in situations where maximum likelihood estimation may be complex or computationally intensive.
Moment Generating Function: The moment generating function (MGF) is a mathematical function used to encapsulate all the moments of a probability distribution, defined as the expected value of the exponential function of a random variable. It connects directly to various aspects such as expected values and variances, making it a powerful tool for analyzing continuous random variables. The MGF can also simplify the process of finding moments and help in determining the distribution of functions of random variables.
Normal Distribution: Normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions. This distribution is vital in various fields due to its properties, such as being defined entirely by its mean and standard deviation, and it forms the basis for statistical methods including hypothesis testing and confidence intervals.
Reliability Analysis: Reliability analysis is a statistical method used to assess the consistency and dependability of a system or component over time. It focuses on determining the probability that a system will perform its intended function without failure during a specified period under stated conditions. This concept is deeply interconnected with random variables and their distributions, as understanding the behavior of these variables is crucial for modeling the reliability of systems and processes.
Signal Processing: Signal processing involves the analysis, interpretation, and manipulation of signals, which can be any physical quantity that varies over time or space. This field is crucial for extracting meaningful information from raw data, enabling the effective transformation and representation of random variables, understanding correlations, and analyzing processes that change over time.
Skewness: Skewness is a statistical measure that describes the asymmetry of a probability distribution around its mean. A positive skew indicates that the right tail of the distribution is longer or fatter than the left, while a negative skew means the left tail is longer or fatter. Understanding skewness helps in analyzing the shape of data distributions and their implications for various higher-order moments and central moments.
Third Moment: The third moment of a random variable is a statistical measure that quantifies the asymmetry of a probability distribution around its mean. It plays a crucial role in understanding the skewness of the distribution, which helps identify whether the data leans towards one side or the other. The third moment is calculated as the expected value of the cube of the deviations from the mean, providing insights into the shape and behavior of data.
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