Density, cumulative distribution function, quantile function and random variate generation for many standard probability distributions are available in the stats package.
The functions for the density/mass function, cumulative distribution
function, quantile function and random variate generation are named in the
For the beta distribution see
For the binomial (including Bernoulli) distribution see
For the Cauchy distribution see
For the chi-squared distribution see
For the exponential distribution see
For the F distribution see
For the gamma distribution see
For the geometric distribution see
dgeom. (This is also
a special case of the negative binomial.)
For the hypergeometric distribution see
For the log-normal distribution see
For the multinomial distribution see
For the negative binomial distribution see
For the normal distribution see
For the Poisson distribution see
For the Student's t distribution see
For the uniform distribution see
For the Weibull distribution see
RNG about random number generation in R.
The CRAN task view on distributions, https://CRAN.R-project.org/view=Distributions, mentioning several CRAN packages for additional distributions.