R Programming/Probability Distributions
This page review the main probability distributions and describe the main R functions to deal with them.
R has lots of probability functions.
r
is the generic prefix for random variable generator such asrunif()
,rnorm()
.d
is the generic prefix for the probability density function such asdunif()
,dnorm()
.p
is the generic prefix for the cumulative density function such aspunif()
,pnorm()
.q
is the generic prefix for the quantile function such asqunif()
,qnorm()
.
Discrete distributions Edit
Benford Distribution Edit
The Benford distribution is the distribution of the first digit of a number. It is due to Benford 1938[1] and Newcomb 1881[2].
> library(VGAM)
> dbenf(c(1:9))
[1] 0.30103000 0.17609126 0.12493874 0.09691001 0.07918125 0.06694679 0.05799195 0.05115252 0.04575749
Bernoulli Edit
We can draw from a Bernoulli using sample(), runif() or rbinom() with size = 1.
> n <- 1000
> x <- sample(c(0,1), n, replace=T)
> x <- sample(c(0,1), n, replace=T, prob=c(0.3,0.7))
> x <- runif(n) > 0.3
> x <- rbinom(n, size=1, prob=0.2)
Binomial Edit
We can sample from a binomial distribution using the rbinom()
function with arguments n
for number of samples to take, size
defining the number of trials and prob
defining the probability of success in each trial.
> x <- rbinom(n=100,size=10,prob=0.5)
Hypergeometric distribution Edit
We can sample n
times from a hypergeometric distribution using the rhyper()
function.
> x <- rhyper(n=1000, 15, 5, 5)
Geometric distribution Edit
> N <- 10000
> x <- rgeom(N, .5)
> x <- rgeom(N, .01)
Multinomial Edit
> sample(1:6, 100, replace=T, prob= rep(1/6,6))
Negative binomial distribution Edit
The negative binomial distribution is the distribution of the number of failures before k successes in a series of Bernoulli events.
> N <- 100000
> x <- rnbinom(N, 10, .25)
Poisson distribution Edit
We can draw n
values from a Poisson distribution with a mean set by the argument lambda
.
> x <- rpois(n=100, lambda=3)
Zipf's law Edit
The distribution of the frequency of words is known as Zipf's Law. It is also a good description of the distribution of city size[3]. dzipf() and pzipf() (VGAM)
> library(VGAM)
> dzipf(x=2, N=1000, s=2)
Continuous distributions Edit
Beta and Dirichlet distributions Edit
- Beta distribution
- Dirichlet in gtools and MCMCpack
>library(gtools)
>?rdirichlet
>library(bayesm)
>?rdirichlet
>library(MCMCpack)
>?Dirichlet
Cauchy Edit
We can sample n
values from a Cauchy distribution with a given location
parameter (default is 0) and scale
parameter (default is 1) using the rcauchy()
function.
> x <- rcauchy(n=100, location=0, scale=1)
Chi Square distribution Edit
Quantile of the Chi-square distribution ( distribution)
> qchisq(.95,1)
[1] 3.841459
> qchisq(.95,10)
[1] 18.30704
> qchisq(.95,100)
[1] 124.3421
Exponential Edit
We can sample n
values from a exponential distribution with a given rate
(default is 1) using the rexp()
function
> x <- rexp(n=100, rate=1)
Fisher-Snedecor Edit
We can draw the density of a Fisher distribution (F-distribution) :
> par(mar=c(3,3,1,1))
> x <- seq(0,5,len=1000)
> plot(range(x),c(0,2),type="n")
> grid()
> lines(x,df(x,df1=1,df2=1),col="black",lwd=3)
> lines(x,df(x,df1=2,df2=1),col="blue",lwd=3)
> lines(x,df(x,df1=5,df2=2),col="green",lwd=3)
> lines(x,df(x,df1=100,df2=1),col="red",lwd=3)
> lines(x,df(x,df1=100,df2=100),col="grey",lwd=3)
> legend(2,1.5,legend=c("n1=1, n2=1","n1=2, n2=1","n1=5, n2=2","n1=100, n2=1","n1=100, n2=100"),col=c("black","blue","green","red","grey"),lwd=3,bty="n")
Gamma Edit
We can sample n
values from a gamma distribution with a given shape
parameter and scale
parameter using the rgamma()
function. Alternatively a shape
parameter and rate
parameter can be given.
> x <- rgamma(n=10, scale=1, shape=0.4)
> x <- rgamma(n=100, scale=1, rate=0.8)
Levy Edit
We can sample n
values from a Levy distribution with a given location parameter (defined by the argument m
, default is 0) and scaling parameter (given by the argument s
, default is 1) using the rlevy()
function.
> x <- rlevy(n=100, m=0, s=1)
Log-normal distribution Edit
We can sample n
values from a log-normal distribution with a given meanlog
(default is 0) and sdlog
(default is 1) using the rlnorm()
function
> x <- rlnorm(n=100, meanlog=0, sdlog=1)
Edit
We can sample n
values from a normal or gaussian Distribution with a given mean
(default is 0) and sd
(default is 1) using the rnorm()
function
> x <- rnorm(n=100, mean=0, sd=1)
Quantile of the normal distribution
> qnorm(.95)
[1] 1.644854
> qnorm(.975)
[1] 1.959964
> qnorm(.99)
[1] 2.326348
- The mvtnorm package includes functions for multivariate normal distributions.
- rmvnorm() generates a multivariate normal distribution.
> library(mvtnorm)
> sig <- matrix(c(1, 0.8, 0.8, 1), 2, 2)
> r <- rmvnorm(1000, sigma = sig)
> cor(r)
[,1] [,2]
[1,] 1.0000000 0.8172368
[2,] 0.8172368 1.0000000
Pareto Distributions Edit
- Generalized Pareto dgpd() in evd
dpareto(), ppareto(), rpareto(), qpareto()
in actuar- The VGAM package also has functions for the Pareto distribution.
Student's t distribution Edit
Quantile of the Student t distribution
> qt(.975,30)
[1] 2.042272
> qt(.975,100)
[1] 1.983972
> qt(.975,1000)
[1] 1.962339
The following lines plot the .975th quantile of the t distribution in function of the degrees of freedom :
curve(qt(.975,x), from = 2 , to = 100, ylab = "Quantile 0.975 ", xlab = "Degrees of freedom", main = "Student t distribution")
abline(h=qnorm(.975), col = 2)
Uniform distribution Edit
We can sample n
values from a uniform distribution (also known as a rectangular distribution] between two values (defaults are 0 and 1) using the runif()
function
> runif(n=100, min=0, max=1)
Weibull Edit
We can sample n
values from a Weibull distribution with a given shape
and scale
parameter (default is 1) using the rweibull()
function.
> x <- rweibull(n=100, shape=0.5, scale=1)
Edit
- The Gumbel distribution
- The logistic distribution : distribution of the difference of two gumbel distributions.
plogis, qlogis, dlogis, rlogis
- Frechet
dfrechet()
evd - Generalized Extreme Value
dgev()
evd - Gumbel
dgumbel()
evd Burr, dburr, pburr, qburr, rburr
in actuar
Distribution in circular statistics Edit
- Functions for circular statistics are included in the CircStats package.
dvm()
Von Mises (also known as the nircular normal or Tikhonov distribution) density functiondtri()
triangular density functiondmixedvm()
Mixed Von Mises densitydwrpcauchy()
wrapped Cauchy densitydwrpnorm()
wrapped normal density.
See also Edit
- Packages VGAM, SuppDists, actuar, fBasics, bayesm, MCMCpack
References Edit
- ↑ Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, 78, 551–572.
- ↑ Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4, 39–40.
- ↑ Gabaix, Xavier (August 1999). "Zipf's Law for Cities: An Explanation". Quarterly Journal of Economics 114 (3): 739–67. doi:10.1162/003355399556133. ISSN 0033-5533. http://pages.stern.nyu.edu/~xgabaix/papers/zipf.pdf.