Lecture on Skewness and Moments

Introduction

• Skewness and moments are interconnected topics.
• Expected value and variance are particular cases of moments.
• Moments provide a quantitative measure of the shape of a function.
• Concept linked to physics, notably density, center of mass, and rotational inertia.

Definition of Moments

• Moments are defined for both discrete and continuous cases:
  • Discrete case: ( \mu_k = \sum (x_i - c)^k f(x_i) )
  • Continuous case: ( \mu_k = \int (x - c)^k f(x) , dx )
• Raw moments occur when ( c = 0 ).
  • 0th raw moment is 1 (probability distribution integral equals 1).
  • First raw moment is the expected value of ( x ).

Central Moments

• Central moments: when ( c = \text{E}(x) ).
  • ( \mu_k = \sum (x_i - \text{E}(x))^k f(x_i) ) (discrete)
  • ( \mu_k = \int (x - \text{E}(x))^k f(x) , dx ) (continuous)
• Variance is the second central moment.

Standardized Moments

• Standardized moments are dimensionless, normalized by standard deviation.
• Third standardized moment: Skewness.
  • Measures asymmetry of a PDF about its mean.
  • Positive skew: right tail longer, mass on left.
  • Negative skew: left tail longer, mass on right.
  • Zero skew: symmetric distribution.

Kurtosis

• Fourth standardized moment: Kurtosis.
  • Describes the thickness of distribution tails.
  • Excess Kurtosis: ( \text{kurt}(x) = \nu_4 - 3 ).
  • Mesokurtic: kurtosis of 0 (normal distribution).
  • Leptokurtic: positive kurtosis, fatter tails.
  • Platykurtic: negative kurtosis, broader tails.

Applications and Importance

• Skewness and kurtosis allow numerical description of distribution shapes.
• Provide comparison to normal distribution.

Summary

• Generalizations of moments help in understanding skewness and kurtosis.
• Definitions and calculations for skewness and kurtosis are provided for practical use.
• Next lecture will continue from this point.