Moment generating function
Definition taught in Random variables and distributions, part of Probability
Moment generating function
| Type | Definition |
| Taught in | Random variables and distributions |
| Part | Probability |
| Worked examples | 2 |
| Questions using it | 1 |
The moment generating function of X is M(s)=\mathbb{E} e^{sX}, for those s at which the expectation exists.
Examples
Exponential moments from the transform
Question page: Exponential moments from the transform
Let X be exponential with rate 1. Use the moment generating function to show that \mathbb{E} X^n=n!.
The moment generating function of the rate-1 exponential law is M(s)=1/(1-s) for s<1. Differentiating n times gives M^{(n)}(s)=n!/(1-s)^{n+1}. The moment formula M^{(n)}(0)=\mathbb{E} X^n therefore yields n!.
Recovering the second moment
Question page: Recovering the second moment
A random variable has expectation 2 and variance 1. Find \mathbb{E} X^2.
\mathbb{E} X^2=\operatorname{Var}(X)+(\mathbb{E} X)^2=1+4=5.
Used in
- Taught in Random variables and distributions, where the derivation and examples are.
- Also used in Joint distributions · Functions of random variables and limit theorems
- Questions that use it (1) — Exponential moments from the transform
- All 27 questions on Random variables and distributions
See also
Expectation · Random variable · Cumulative distribution function · Discrete distribution · Continuous distribution · Expectation of a function · Variance · Probability generating function