Moments and Moment Generating Function of the Gamma Distribution
Raw moments, limited moments and moment generating function for the
Gamma distribution with parameters shape and scale.
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)
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 |
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.
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.
Vincent Goulet vincent.goulet@act.ulaval.ca, Christophe Dutang and Mathieu Pigeon
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.
mgamma(2, 3, 4) - mgamma(1, 3, 4)^2 levgamma(10, 3, 4, order = 2) mgfgamma(1,3,2)
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