# GammaSupp: Moments and Moment Generating Function of the Gamma... In actuar: Actuarial Functions and Heavy Tailed Distributions

 GammaSupp R Documentation

## Moments and Moment Generating Function of the Gamma Distribution

### Description

Raw moments, limited moments and moment generating function for the Gamma distribution with parameters `shape` and `scale`.

### Usage

```mgamma(order, shape, rate = 1, scale = 1/rate)
levgamma(limit, shape, rate = 1, scale = 1/rate, order = 1)
mgfgamma(t, shape, rate = 1, scale = 1/rate, log = FALSE)
```

### Arguments

 `order` order of the moment. `limit` limit of the loss variable. `rate` an alternative way to specify the scale. `shape, scale` shape and scale parameters. Must be strictly positive. `t` numeric vector. `log` logical; if `TRUE`, the cumulant generating function is returned.

### Details

The kth raw moment of the random variable X is E[X^k], the kth limited moment at some limit d is E[min(X, d)^k] and the moment generating function is E[e^{tX}], k > -shape.

### Value

`mgamma` gives the kth raw moment, `levgamma` gives the kth moment of the limited loss variable, and `mgfgamma` gives the moment generating function in `t`.

Invalid arguments will result in return value `NaN`, with a warning.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca, Christophe Dutang and Mathieu Pigeon

### References

Johnson, N. L. and Kotz, S. (1970), Continuous Univariate Distributions, Volume 1, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

`GammaDist`
```mgamma(2, 3, 4) - mgamma(1, 3, 4)^2