gamma {distributionsrd}

The Gamma distribution

Description

Raw moments for the Gamma distribution.

Usage

mgamma(
  r = 0,
  truncation = 0,
  shape = 2,
  rate = 1,
  scale = 1/rate,
  lower.tail = TRUE
)

Arguments

r

rth raw moment of the distribution, defaults to 1.

truncation

lower truncation parameter, defaults to 0.

shape, rate, scale

shape, rate and scale of the distribution with default values of 2 and 1 respectively.

lower.tail

logical; if TRUE (default), moments are E[x^r|X \le y], otherwise, E[x^r|X > y]

Details

Probability and Cumulative Distribution Function:

f(x) = \frac{1}{s^k\Gamma(k)}\omega^{k-1}e^{-\frac{\omega}{s}},\qquad F_X(x) = \frac{1}{\Gamma(k)}\gamma(k,\frac{\omega}{s})

,

where \Gamma(x) stands for the upper incomplete gamma function function, while \gamma(s,x) stands for the lower incomplete Gamma function with upper bound x.

The y-bounded r-th raw moment of the distribution equals:

\mu^r_y = \frac{s^{r}}{\Gamma(k)} \Gamma\left(r + k , \frac{y}{s} \right)

Value

Provides the truncated rth raw moment of the distribution.

## The zeroth truncated moment is equivalent to the probability function pgamma(2,shape=2,rate=1) mgamma(truncation=2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample, #for large enough samples. x = rgamma(1e5,shape=2,rate=1) mean(x) mgamma(r=1,lower.tail=FALSE)

sum(x[x>quantile(x,0.1)])/length(x) mgamma(r=1,truncation=quantile(x,0.1),lower.tail=FALSE)