# InvGamma: Inverse-gamma distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

Density, distribution function and random generation for the inverse-gamma distribution.

## Usage

 ```1 2 3 4 5 6 7``` ```dinvgamma(x, alpha, beta = 1, log = FALSE) pinvgamma(q, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE) qinvgamma(p, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE) rinvgamma(n, alpha, beta = 1) ```

## Arguments

 `x, q` vector of quantiles. `alpha, beta` positive valued shape and scale parameters. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

## Details

Probability mass function

f(x) = (β^α * x^(-α-1) * exp(-β/x)) / Γ(α)

Cumulative distribution function

F(x) = γ(α, β/x) / Γ(α)

## References

Witkovsky, V. (2001). Computing the distribution of a linear combination of inverted gamma variables. Kybernetika 37(1), 79-90.

Leemis, L.M. and McQueston, L.T. (2008). Univariate Distribution Relationships. American Statistician 62(1): 45-53.

`GammaDist`
 ```1 2 3 4 5 6``` ```x <- rinvgamma(1e5, 20, 3) hist(x, 100, freq = FALSE) curve(dinvgamma(x, 20, 3), 0, 1, col = "red", add = TRUE, n = 5000) hist(pinvgamma(x, 20, 3)) plot(ecdf(x)) curve(pinvgamma(x, 20, 3), 0, 1, col = "red", lwd = 2, add = TRUE, n = 5000) ```