MGF_gamma.RdMoment Generating Function (MGF) of the Gamma distribution with shape parameter \(\alpha\) and rate parameter \(\beta\).
MGF_gamma(t, shape, rate = 1/scale, scale = 1/rate)
| t | t. |
|---|---|
| shape | shape parameter \(\alpha\), must be positive integer. |
| rate | \(\beta\) is the rate parameter, must be positive. |
| scale | alternative parameterization to rate parameter, scale = 1 / rate. |
Function :
MGF_gamma gives the moment generating function (MGF).
E_gamma gives the expected value.
V_gamma gives the variance.
kthmoment_gamma gives the kth moment.
Etrunc_gamma gives the truncated mean.
SL_gamma gives the stop-loss.
Elim_gamma gives the limited mean.
Mexcess_gamma gives the mean excess loss.
TVaR_gamma gives the Tail Value-at-Risk.
Invalid parameter values will return an error detailing which parameter is problematic.
The Gamma distribution with shape parameter \(\alpha\) and rate parameter \(\beta\) has density: $$f\left(x\right) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1}% \textrm{e}^{-\beta x}$$ for \(x \in \mathcal{R}^+\), \(\beta, \alpha > 0\).
Other Gamma Distribution:
E_gamma(),
Elim_gamma(),
Etrunc_gamma(),
Mexcess_gamma(),
SL_gamma(),
TVaR_gamma(),
V_gamma(),
VaR_gamma(),
kthmoment_gamma()
MGF_gamma(t = 1, shape = 3, rate = 5)#> [1] 1.953125