Editing Moment-generating function (section)

==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 <math>M_X(t)</math> when the latter exists.
{|class="wikitable" style="padding-left:1.5em;"
|-
! Distribution
! Moment-generating function <math>M_X(t)</math>
! Characteristic function <math>\varphi (t)</math>
|-
|[[Degenerate distribution|Degenerate]] <math>\delta_a</math>
|<math>e^{ta}</math>
|<math>e^{ita}</math>
|-
| [[Bernoulli distribution|Bernoulli]] <math>P(X = 1) = p</math> 
| <math>1 - p + pe^t</math>
| <math>1 - p + pe^{it}</math>
|-
| [[Binomial distribution|Binomial]] <math>B(n, p)</math>
| <math>\left(1 - p + pe^t\right)^n</math>
| <math>\left(1 - p + pe^{it}\right)^n</math>
|-
| [[Geometric distribution|Geometric]]  <math>(1 - p)^{k}\,p</math>
| <math>\frac{p}{1 - (1 - p) e^t}, ~ t < -\ln(1 - p)</math>
| <math>\frac{p}{1 - (1 - p)\,e^{it}}</math>
|-
|[[Negative binomial distribution|Negative binomial]] <math>\operatorname{NB}(r, p)</math>
|<math>\left(\frac{p}{1 - e^t + pe^t}\right)^r, ~ t<-\ln(1-p)</math>
|<math>\left(\frac{p}{1 - e^{it} + pe^{it}}\right)^r</math>
|-
| [[Poisson distribution|Poisson]] <math>\operatorname{Pois}(\lambda)</math>
| <math>e^{\lambda(e^t - 1)}</math> 
| <math>e^{\lambda(e^{it} - 1)}</math> 
|- 
| [[Uniform distribution (continuous)|Uniform (continuous)]] <math>\operatorname U(a, b)</math>
| <math>\frac{e^{tb} - e^{ta}}{t(b - a)}</math>
| <math>\frac{e^{itb} - e^{ita}}{it(b - a)}</math>
|- 
| [[Discrete uniform distribution|Uniform (discrete)]] <math>\operatorname{DU}(a, b)</math>
| <math>\frac{e^{at} - e^{(b + 1)t}}{(b - a + 1)(1 - e^t)}</math>
| <math>\frac{e^{ait} - e^{(b + 1)it}}{(b - a + 1)(1 - e^{it})}</math>
|-
|[[Laplace distribution|Laplace]] <math>L(\mu, b)</math>
|<math>\frac{e^{t\mu}}{1 - b^2t^2}, ~ |t| < 1/b</math>
|<math>\frac{e^{it\mu}}{1 + b^2t^2}</math>
|-
| [[Normal distribution|Normal]] <math>N(\mu, \sigma^2)</math>
| <math>e^{t\mu + \sigma^2 t^2 / 2}</math>
| <math>e^{it\mu - \sigma^2 t^2 / 2}</math>
|-
| [[Chi-squared distribution|Chi-squared]] <math>\chi^2_k</math>
| <math>{\left(1 - 2t\right)}^{-k/2}, ~ t < 1/2</math>
| <math>{\left(1 - 2it\right)}^{-{k}/{2}}</math>
|-
|[[Noncentral chi-squared distribution|Noncentral chi-squared]] <math>\chi^2_k(\lambda)</math>
| <math>e^{\lambda t/(1-2t)} {\left(1 - 2t\right)}^{-k/2}</math>
| <math>e^{i\lambda t/(1-2it)} {\left(1 - 2it\right)}^{-k/2}</math>
|-
| [[Gamma distribution|Gamma]] <math>\Gamma(k, \tfrac{1}{\theta})</math>
|<math>{\left(1 - t\theta\right)}^{-k}, ~ t < \tfrac{1}{\theta}</math>
| <math>{\left(1 - it\theta\right)}^{-k}</math>
|-
| [[Exponential distribution|Exponential]] <math>\operatorname{Exp}(\lambda)</math>
| <math>\left(1 - t\lambda^{-1}\right)^{-1}, ~ t < \lambda</math>
| <math>\left(1 - it\lambda^{-1}\right)^{-1}</math>
|-
|[[Beta distribution|Beta]]
|<math>1  +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}</math>
|<math>{}_1F_1(\alpha; \alpha+\beta; i\,t)\! </math> (see [[Confluent hypergeometric function]])
|-
| [[Multivariate normal distribution|Multivariate normal]] <math>N(\mathbf{\mu}, \mathbf{\Sigma})</math>
|<math>\exp\left[\mathbf{t}^\mathrm{T} \left(  \boldsymbol{\mu} + \tfrac{1}{2} \boldsymbol{\Sigma} \mathbf{t}\right)\right]</math>
|<math>\exp\left[\mathbf{t}^\mathrm{T} \left(i \boldsymbol{\mu} - \tfrac{1}{2} \boldsymbol{\Sigma} \mathbf{t}\right)\right]</math>
|-
| [[Cauchy distribution|Cauchy]] <math>\operatorname{Cauchy}(\mu, \theta)</math>
|[[Indeterminate form|Does not exist]]
| <math>e^{it\mu - \theta|t|}</math>
|-
|[[Multivariate Cauchy distribution|Multivariate Cauchy]] 
<math>\operatorname{MultiCauchy}(\mu, \Sigma)</math><ref>Kotz et al.{{full citation needed|date=December 2019}} p. 37 using 1 as the number of degree of freedom to recover the Cauchy distribution</ref>
|Does not exist
|<math>\exp\left(i\mathbf{t}^{\mathrm{T}}\boldsymbol\mu - \sqrt{\mathbf{t}^{\mathrm{T}}\boldsymbol{\Sigma} \mathbf{t}}\right)</math>
|}