Expectation of Random Variables

Expectation and Moments

Definition

The expected value, or mean, or first moment of a random variable is defined to be

Assuming that the sum (or integral) is well defined. We use the following notation to denote the expected value of :

Theorem (Rule of the Lazy Statistician)

Let . Then

Let be an event, and let where if and if . Then,

In other words, probability is a special case of expectation.

Definition

The th moment of is defined to be assuming that .

Theorem

If the th moment exists, and if , then the th moment exists.

Definition

The th central moment is defined to be .

Properties of Expectation

Theorem

If are random variables, and are constants, then

Theorem

Let be independent random variables. Then,

Notice that the summation rule does not require independence, but the multiplication rule does.

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 3.1.