What Is a Moment?

For a random variable $X$, its moments are simply the expected values of its powers:

$$\mathbb{E}(X^n).$$

We can write the $n$th moment as

$$M_n=\mathbb{E}(X^n).$$

Notice that $M_n$ is just a number. It is not itself a random variable or a function.

The First and Second Moments

The first moment is

$$M_1=\mathbb{E}(X).$$

This is simply the mean of $X$, which we often write as $\mu$:

$$M_1=\mathbb{E}(X)=\mu.$$

The second moment is

$$M_2=\mathbb{E}(X^2).$$

The first two moments are especially important because together they give us the variance:

$$\operatorname{Var}(X)=\mathbb{E}(X^2)-(\mathbb{E}(X))^2=M_2-M_1^2.$$

In probability, statistics, and econometrics, these are the moments we will use most often.

Higher Moments

We can continue in exactly the same way.

The third moment is

$$M_3=\mathbb{E}(X^3),$$

the fourth moment is

$$M_4=\mathbb{E}(X^4),$$

and, for example, the sixth moment is

$$M_6=\mathbb{E}(X^6).$$

Higher moments can also tell us more about the shape of a distribution. In particular, versions of the third and fourth moments that are centred around the mean and suitably scaled are used to measure skewness and kurtosis. Roughly speaking, skewness describes the asymmetry of a distribution, while kurtosis is related to how much probability weight lies in the tails.

As well as finding the expected values directly, another way to find moments is by using a “moment generating function”. This is covered on the next page.

Background:

• Expected Values

• Variance

Extensions:

• Moment Generating Functions