Description Usage Arguments Details Value
Raw moments for the Gamma distribution.
1 2 3 4 5 6 7 8 |
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 ≤ y], otherwise, E[x^r|X > y] |
Probability and Cumulative Distribution Function:
f(x) = \frac{1}{s^kΓ(k)}ω^{k-1}e^{-\frac{ω}{s}},\qquad F_X(x) = \frac{1}{Γ(k)}γ(k,\frac{ω}{s})
,
where Γ(x) stands for the upper incomplete gamma function function, while γ(s,x) stands for the lower incomplete Gamma function with upper bound x.
The y-bounded r-th raw moment of the distribution equals:
μ^r_y = \frac{s^{r}}{Γ(k)} Γ≤ft(r + k , \frac{y}{s} \right)
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)
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