Moment-Generating Functions: Definition, Equations & Examples

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  • 0:04 Moments
  • 0:43 Moment-Generating Functions
  • 1:24 Finding Expected Value…
  • 2:05 Example Problems
  • 4:05 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson, you'll discover what moment-generating functions are and how we use them to find moments. You'll then go on to explore the relationship between moment-generating functions and the expected value and variance of a probability distribution.


When studying random variables and their probability distributions, a couple of the early concepts a statistics student learns are expected value and variance. The expected value or mean of a random variable (X) is its average, and variance is the spread of the probability distribution.

Expected value and variance are both examples of quantities known as moments, where moments are used to make measurements about the central tendency of a set of values. We can find the moments of a probability distribution using its moment-generating function. In this lesson, we will learn how to find a moment-generating function, as well as how to use it to find expected value and variance.

Moment-Generating Functions

A moment-generating function, or MGF, as its name implies, is a function used to find the moments of a given random variable. The formula for finding the MGF (M(t)) is as follows, where E is expected value:

mgf general formula

This is the general formula for finding an MGF under the condition that there exists a positive number b, such that b ≤ |t|. How we go about calculating this in practice differs depending on whether we're working with a discrete or continuous probability distribution.

mgf for discrete and continuous probability distributions

In this equation, p(x) and f(x) are the density functions of their given probability distributions.

Finding Expected Value and Variance

Once we have an MGF, we need to know how to use it to generate moments. We do this by taking derivatives of the MGF and evaluating it at t equals 0.

moment formula

Every consecutive derivative of the MGF gives you a different moment. Each moment is equal to the expected value of X raised to the power of the number of the moment.

By taking the first derivative (n = 1) of the MGF and setting t equal to 0, we find the expected value or mean of random variable X. The second derivative (n = 2) then gives us the expected value of X2, which can be used to find variance with the following formula:

variance formula

Example Problems

In order to fully grasp how we use the MGF, let's work through a couple of problems together. For our first problem, we'll find the MGF for a known probability distribution. In this case, let's find the MGF of the binomial distribution. If you need a quick reminder, the binomial distribution is a discrete probability distribution, and its density function is given below, where p is the probability of success and q = 1 - p:

binomial distribution density function

To find this MGF, we're going to have to work with manipulating a series denoted by the sum sign (∑), as you'll have to do when finding the MGF of a discrete probability distribution. If you're a little hazy on dealing with these, it might be worth your time to review some common ways we work with series in math classes, such as the Taylor series or the binomial theorem. In fact, we'll need the binomial theorem to be able to solve this problem.

binomial theorem

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