Examples
Here are some examples of the moment generating function and the characteristic function for comparison. It can be seen that the characteristic function is a Wick rotation of the moment generating function Mx(t) when the latter exists.
| Distribution | Moment-generating function MX(t) | Characteristic function φ(t) |
|---|---|---|
| Bernoulli | ||
| Geometric | , for |
|
| Binomial B(n, p) | ||
| Poisson Pois(λ) | ||
| Uniform (continuous) U(a, b) | ||
| Uniform (discrete) U(a, b) | ||
| Normal N(μ, σ2) | ||
| Chi-squared χ2k | ||
| Gamma Γ(k, θ) | ||
| Exponential Exp(λ) | ||
| Multivariate normal N(μ, Σ) | ||
| Degenerate δa | ||
| Laplace L(μ, b) | ||
| Negative Binomial NB(r, p) | ||
| Cauchy Cauchy(μ, θ) | does not exist |
Read more about this topic: Moment-generating Function
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)