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
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