Introduction

In the world of probability and statistics, moments play a crucial role in describing the characteristics of a random variable’s distribution. Moments provide insight into the central tendencies, variability, and shape of the distribution. In this chapter, we will delve into the concept of population moments, specifically focusing on the four common moments that help us understand various aspects of a distribution.

Population Moments Defined

Population moments are numerical values that summarize the distribution of a random variable. These moments provide valuable information about the shape, location, and spread of the distribution. There are several moments, but we will concentrate on the first four moments, which are fundamental in statistical analysis.

First Moment - Mean

The first moment, also known as the mean, is a measure of the central tendency of a distribution. Mathematically, the mean $(\mu)$ of a random variable $X$ is calculated as:

$$\mu=E[X]=\sum_i x_i \cdot P\left(X=x_i\right)$$

Second Moment - Variance

The second moment, called the variance, quantifies the dispersion or spread of a distribution. It is calculated as the average of the squared differences between each value and the mean. The formula for variance $\left(\sigma^{\wedge} 2\right)$ is given by:

$$\sigma^2=E\left[(X-\mu)^2\right]=\sum_i\left(x_i-\mu\right)^2 \cdot P\left(X=x_i\right)$$

Third Moment - Skewness

Skewness measures the asymmetry of a distribution. A positively skewed distribution has a longer tail on the right, while a negatively skewed distribution has a longer tail on the left. The skewness coefficient $(\gamma)$ is calculated using the formula:

$$\gamma=\frac{E\left[(X-\mu)^3\right]}{\sigma^3}=\frac{\sum_i\left(x_i-\mu\right)^3 \cdot P\left(X=x_i\right)}{\sigma^3}$$

Fourth Moment - Kurtosis

Kurtosis indicates the degree of heaviness of the tails of a distribution compared to a normal distribution. It reveals information about the presence of outliers. The formula for kurtosis $(\kappa)$ is:

$$\kappa=\frac{E\left[(X-\mu)^4\right]}{\sigma^4}=\frac{\sum_i\left(x_i-\mu\right)^4 \cdot P\left(X=x_i\right)}{\sigma^4}$$