11.1.2. Types of Moments

1. Raw Moments (or Moments about the Origin) The r-th raw moment of a random variable 𝑋 is defined as:

𝜇′ = 𝐸[𝑋𝑟]

where 𝐸 denotes the expectation.

Raw moments provide a way to quantify characteristics of a random variable based on its original value. Specifically, the r-th raw moment is calculated as the expected value of the random variable raised to the power of r. This means, for example, that the first raw moment (when r=1) represents the mean of the distribution, while the second raw moment (r=2) accounts for the average of the squared values, which is related to variance.

Imagine you want to understand how high students jump in a gym class. If you measure each jump and average those heights, you get the first raw moment, which tells you the average jump height. If you square each jump height before averaging, you’re calculating the second raw moment, giving you insights about the variability or spread of jump heights.

2. Central Moments The r-th central moment is the expected value of the r-th power of deviations from the mean:

𝜇 = 𝐸[(𝑋−𝜇)𝑟]

where 𝜇 = 𝐸[𝑋] is the mean of the distribution.

Central moments focus on how the values of a random variable deviate from its mean. The r-th central moment measures the average of the r-th power of these deviations. This concept is important because it helps to standardize the moments around the mean, rather than the origin. For instance, the second central moment provides the variance of the distribution, which tells us how spread out the data points are around the mean.

Consider analyzing the scores of a class on a math test. The mean score gives you a central value, but to understand how consistent or varied those scores are, you look at how far each score is from that mean (the deviations). If you square these deviations and average them (second central moment), you get the variance, which helps you see if most students scored similarly or if there were large discrepancies.

Important Moments Moment Order Name Formula Significance 1st Mean 𝜇 = 𝐸[𝑋] Measures the central tendency 2nd Variance 𝜎2 = 𝐸[(𝑋− 𝜇)2] Measures spread or dispersion 3rd Skewness 𝜎3 Measures asymmetry of distribution 4th Kurtosis 𝜎4 Measures peakedness or flatness.

This section identifies key types of moments known as �important moments�, each with its significance. The first moment is the mean, which indicates the average value of the distribution. The second moment is variance, reflecting how data points are spread out. The third moment measures skewness, informing us about the symmetry of the distribution—whether it leans more to the left or right. Finally, the fourth moment, kurtosis, tells us about the shape of the distribution—specifically its 'peakedness' and how heavy the tails are.

If you think of moments in terms of a sports game, the mean could represent the average number of points scored by a team, variance reflects how much scores vary from game to game, skewness tells whether one team consistently beats the other by a lot or just a little, and kurtosis shows whether scores are consistently close to the average or there are a few very high scoring games that pull the average up.

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