Statistics Definitions > Moment Generating Function (MGF)

If you aren’t familiar with moments, you may want to read this article first: what is a moment?

**Contents (Click to skip to that section):**

## 1. What is a Moment Generating Function?

**Moment generating functions** are a way to find moments like the mean(μ) and the variance(σ^{2}). They are an alternative way to represent a probability distribution with a simple one-variable function.

**Each probability distribution has a unique MGF**, which means they are especially useful for solving problems like finding the distribution for sums of random variables. They can also be used as a proof of the Central Limit Theorem.

There isn’t an intuitive definition for exactly what an MGF *is*; it’s merely **a computational tool**.

## How to Find an MGF

Finding an MGF for a discrete random variable involves summation; for continuous random variables, calculus is used. It’s actually very simple to create moment generating functions if you are comfortable with summation and/or differentiation and integration:

For the above formulas, f(x) is the probability density function of X and the integration range (listed as -∞ to ∞) will change depending on what range your function is defined for.

**Example**: Find the MGF for e^{-x}.

**Solution**:

Step 1: Plug e^{-x} in for fx(x) to get:

Note that I changed the lower integral bound to zero, because this function is only valid for values higher than zero.

Step 2: Integrate. The MGF is 1 / (1-t).

The moment generating function only works when the integral converges on a particular number. The above integral *diverges *(spreads out) for t values of 1 or more, so the MGF only exists for values of t less than 1. You’ll find that most continuous distributions aren’t defined for larger values (say, above 1). This is usually *not* an issue: in order to find expected values and variances, the MGF only needs to be found for small t values close to zero.

## Using the MGF

Once you’ve found the moment generating function, you can use it to find expected value, variance, and other moments.

- M(0) = 1,
- M′(0) = E(X),
- M′′(0) = E(X
^{2}), - M′′′(0) = E(X
^{3})

and so on;

Var(X) = M′′(0) − M′(0)^{2}.

**Example:** Find E(X^{3}) using the MGF (1-2t)^{-10}.

Step 1: Find the third derivative of the function (the list above defines M′′′(0) as being equal to E(X^{3}); before you can evaluate the derivative at 0, you first need to find it):

M′′′(t) = (−2)^{3}(−10)(−11)(−12)(1 − 2^{t})^{-13}

Step 2: Evaluate the derivative at 0:

M′′′(0) = (−2)^{3}(−10)(−11)(−12)(1 − 2^{t})^{-13}

= (−2)^{3}(−10)(−11)(−12)(1)

= 10,560.

**Solution**: E(X^{3}) = 10,560.

## What is a Probability Generating Function?

A **probability generating function** contains the same information as a moment generating function, with one important difference: the probability generating function is normally used for **non-negative integer valued** random variables.

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