Extension: Nth Central Moment of a Random Variable

$Nth$ Central Moment of a Random Variable

The nth central moment is a number that helps us to describe the shape of a probability distribution. It is based on how far each value is from the mean. You may already know that the mean tells us the center of a data set or random variable. The first step in finding a central moment is to find this mean.

For a random variable $X$, the mean is written as $\mu$. We then look at the difference between each possible value of $X$ and the mean. This difference is written as $X-\mu$. We raise this difference to a power, called $n$. The $nth$ central moment is the expected value of this result.

####The Formula

The formula is:

$$\mu_n = E[(X-\mu)^n]$$

Here, $\mu_n$ represents the nth central moment. The letter $n$ tells us which moment we are finding. For example, the second central moment uses $n=2$. The second central moment is especially important because it is equal to the variance of the random variable.

The third central moment uses $n=3$. It can help describe whether a distribution is more stretched to one side than the other. This idea is called skewness. Higher central moments can give us even more information about the shape of a distribution.

To find a central moment, we first find the mean. Next, we subtract the mean from each possible value. Then, we raise each difference to the nth power. Finally, we find the expected value of those results.

Central moments connect several ideas from probability and statistics. They build on expected value, mean, and variance. Later, you can use central moments to study more detailed features of probability distributions. This makes them useful tools for understanding the shape and behavior of random variables.

Nth Central Moment

An $nth$ central moment describes how the values of a random variable are spread around their mean.

To find it, we first need to find the mean, represented by $\mu$. Then, we find how far each value $X$ is from the mean using $X-\mu$. This distance tells us whether a value is above or below the mean.

Next, we raise each distance to the $nth$ power. Finally, we use the probabilities of the values to find the expected value of these powered distances.

The formula for the $nth$ central moment is:

$$E(X-\mu)^n]$$

Different values of $n$ give us different ways to describe the random variable. The 2nd moment tells us the variance, the 3rd moment tell us the skewness, and the 4th moment is related to kurtosis.

(Video) nth Central Moment by: Anusha Rahman

In this video by Anusha Rahman, you will learn the basics of the nth central moments.

Summary

A central moment describes the distribution of a random variable by looking at how its values vary around the mean, written as $\mu$. The second central moment is the variance, which is related to standard deviation. Higher order central moments give more information about the shape of the distribution.

The third central moment relates to skewness, while the fourth central moment relates to kurtosis.

Central moments are one set of moments, alongside ordinary moments. Unlike ordinary moments, central moments relate only to how the random variable varies about its mean.

Central Moments

by Mike Dabkowski

In this lesson by Mike Dabkowski, you will learn about the raw and central moments of a random variable.

Summary

This video explains the relationship between raw moments and central moments of a random variable.

It begins by defining the $k^{th}$ raw moment as the expected value of $X^k$. The video also explains that the first raw moment is the mean of the random variable. Next, it defines the $k^{th}$ central moment as the expected value of $X-\mu^k$.

You will see how the second central moment can be written using the first and second raw moments. This leads to the formula for the variance, which is the second central moment. Finally, the video uses the third central moment to show how central moments can be written in terms of powers of raw moments.