gamma function

The Gamma distribution

Raw moments for the Gamma distribution.

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, $E[x^r|X > y]$

Returns

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)

Details

Probability and Cumulative Distribution Function:

,

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: