# Distributions in R and S-PLUS

This page summarizes how to work with univariate probability distributions in R and S-PLUS. See also notes on working with distributions in Mathematica, Excel, and in Python with SciPy.

R and S-PLUS use prefixes and bases to denote functions related to a
distribution. The prefixes are `d`

, `p`

, `q`

,
and `r`

. The bases are the name of the distribution family such
as `norm`

for the normal distribution.

The prefix `d`

is for density, i.e. PDF.

The prefix `p`

is for CDF (cumulative density function), unless the
argument `lower.tail = FALSE`

is supplied, in which case it turns
into the CCDF (complementary CDF).

The prefix `q`

is for the CDF inverse, unless the argument ```
lower.tail = FALSE
```

is

supplied, in which case it turns into the CCDF inverse.

The prefix `r`

is for random sample.

The first argument to a distribution-related function is the ostensible argument. Next come the distribution parameters followed by other options.

## Examples

`pnorm(0.77, 0, 2.1)`

computes F_{X}(0.77) where X is a normal random
variable with mean 0 and standard deviation 2.1 and F_{X} is its CDF.

`dbeta(0.7, 2.1, 3.4)`

computes f_{X}(0.7) where X is
a beta random variable with parameters 2.1 and 3.4 and f_{X }is its
PDF.

`qgamma(0.1, 3.1, 1.0, lower.tail = FALSE)`

finds a value y so
that P(Y > y) = 0.1 where Y has a gamma distribution with shape 3.1 and
scale 1.

## Distributions and parameterizations

Distribution |
Base name |
Parameters |

beta | `beta` |
`shape1` , `shape2` |

binomial | `binom` |
`size` , `prob` |

Cauchy | `cauchy` |
`location` , `scale` |

chi-squared | `chisq` |
`df` |

exponential | `exp` |
`rate` |

F | `f` |
`df1` , `df2` |

gamma | `gamma` |
`shape` , `rate` |

geometric | `geom` |
`p` |

hypergeometric | `hyper` |
`m` , `n` , `k` |

log-normal | `lnorm` |
`meanlog` , `sdlog` |

logistic | `logis` |
`location` , `scale` |

negative binomial | `nbinom` |
`size` , `prob` |

normal | `norm` |
`mean` , `sd` |

Poisson | `pois` |
`lambda` |

Student t | `t` |
`df` |

uniform | `unif` |
`min` , `max` |

Weibull | `weibull` |
`shape` , `scale` |

Note that the exponential is parameterized in terms of the rate, the reciprocal of the mean.

The gamma can be parameterized by its shape and either the rate or the scale. The rate is the default argument by position, but you can specify the scale by name.

The hypergeometric distribution gives the probability of various numbers of red balls when k balls are taken from an urn containing m red balls and n blue balls. Note that another popular convention uses the number of red balls and the total number of balls m+n.

Note that the parameters for the log-normal are the mean and standard deviation of the log of the distribution, not the mean and standard deviation of the distribution itself.