Raw Moment

probability-theory

Definition

Raw Moment

A raw moment (or simply, a moment) is calculated about the origin (zero). The $k$-th raw moment of a random variable $X$ is given by:

$$\mathbb{E}[X^k]$$

Note that for $k=1$, the raw moment is exactly the expected value. That’s why the expected value is called first raw moment.

Intuition

The most fundamental question one can ask is about a distribution is “What is the typical value?” or “Where is its centre?“.

To find this, one can intuitively calculate the average, which we denote as:

$$\mathbb{E}[X]$$

You sum up all values and divide by the count. In the context of a random variable, this becomes the weighted average, where each possible value $x$ is weighted by its probability $f(x

This average acts like the centre of mass or the balance point of the distribution. Thus, the first raw moment answers the most basic question about the distribution’s location.

What happens if $k\to \infty$?:

• Low-order moments describe the “bulk” of the distributions, i.e. is centre and spread.
• High-order moments are almost entirely dominated by the most extreme, rare values in the tails. They tell you about the probability of outliers, not he finite details in the centre.

The next logical question is: “How spread out are the data points from this centre?“.