```
if(!requireNamespace("distributions3")) {
if(interactive() || is.na(Sys.getenv("_R_CHECK_PACKAGE_NAME_", NA))) {
stop("not all packages required for the example are installed")
} else q() }
## package and random seed
library("distributions3")
set.seed(6020)
## three Weibull distributions
X <- GAMLSS("WEI", mu = c(1, 1, 2), sigma = c(1, 2, 2))
X
## moments
mean(X)
variance(X)
## support interval (minimum and maximum)
support(X)
is_discrete(X)
is_continuous(X)
## simulate random variables
random(X, 5)
## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ], main = "WEI(1,1)")
hist(x[2, ], main = "WEI(1,2)")
hist(x[3, ], main = "WEI(2,2)")
## probability density function (PDF) and log-density (or log-likelihood)
x <- c(2, 2, 1)
pdf(X, x)
pdf(X, x, log = TRUE)
log_pdf(X, x)
## cumulative distribution function (CDF)
cdf(X, x)
## quantiles
quantile(X, 0.5)
## cdf() and quantile() are inverses
cdf(X, quantile(X, 0.5))
quantile(X, cdf(X, 1))
## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)
quantile(X, p, elementwise = TRUE)
quantile(X, p, elementwise = TRUE, drop = FALSE)
## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
"theoretical" = mean(X),
"empirical" = rowMeans(random(X, 1000))
)
```

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