Calculus How To

Moment Generating Function MGF: Definition, Examples

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Types of Functions > Moment Generating Function (MGF)

If you aren’t familiar with moments, you may want to read this article first: What are moments?

Contents (Click to skip to that section):

  1. Moment Generating Function
  2. Probability Generating Function

1. What is a Moment Generating Function?

Moment generating functions are a way to find moments like the mean(μ) and the variance2). 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 just a computational tool. Think of it as a formula, in the same way that y = mx + b allows you to create linear functions, the MGF formula helps you to find moments.

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:
moment generating function 2

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.

Step 1: Plug e-x in for fx(x) to get:
moment generating function 3

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(X2),
  • M′′′(0) = E(X3)

and so on;
Var(X) = M′′(0) − M′(0)2.

Example: Find E(X3) 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(X3); before you can evaluate the derivative at 0, you first need to find it):
M′′′(t) = (−2)3(−10)(−11)(−12)(1 − 2t)-13

Step 2: Evaluate the derivative at 0:
M′′′(0) = (−2)3(−10)(−11)(−12)(1 − 2t)-13
= (−2)3(−10)(−11)(−12)(1)
= 10,560.

Solution: E(X3) = 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.

Stephanie Glen. "Moment Generating Function MGF: Definition, Examples" From Calculus for the rest of us!

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