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 as runif(), rnorm(). d is the generic prefix for the probability density function such as dunif(), dnorm(). p is the generic prefix for the cumulative density function such as punif(), pnorm(). q is the generic prefix for the quantile function such as qunif(), qnorm(). Discrete distributions Benford Distribution The Benford distribution is the distribution of the first digit of a number. It is due to Benford 1938Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, 78, 551–572. and Newcomb 1881Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4, 39–40.. > 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 We can draw from a Bernoulli_distribution 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 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 We can sample n times from a hypergeometric distribution using the rhyper() function. > x <- rhyper(n=1000, 15, 5, 5) Geometric distribution The geometric distribution. > N <- 10000 > x <- rgeom(N, .5) > x <- rgeom(N, .01) Multinomial The multinomial distribution. > sample(1:6, 100, replace=T, prob= rep(1/6,6)) Negative binomial distribution 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 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 The distribution of the frequency of words is known as Zipf's Law. It is also a good description of the distribution of city sizeGabaix, 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.. dzipf() and pzipf() (VGAM) > library(VGAM) > dzipf(x=2, N=1000, s=2) Continuous distributions Beta and Dirichlet distributions Beta distribution Dirichlet distribution Dirichlet in gtools and MCMCpack >library(gtools) >?rdirichlet >library(bayesm) >?rdirichlet >library(MCMCpack) >?Dirichlet Cauchy 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 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 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 We can draw the density of a F-distribution 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 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 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 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) Normal and related distributions We can sample n values from a Normal_distribution 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 Pareto_distribution 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 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 We can sample n values from a Uniform_distribution_(continuous) 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 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) Extreme values and related distribution 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 Functions for circular statistics are included in the CircStats package. dvm() Von_Mises_distribution Von Mises (also known as the nircular normal or Tikhonov distribution) density function dtri() Triangular_distribution triangular density function dmixedvm() Mixed Von Mises density dwrpcauchy() wrapped Cauchy density dwrpnorm() wrapped normal density. See also Packages VGAM, SuppDists, actuar, fBasics, bayesm, MCMCpack References Optimization Random Number Generation pt:R (linguagem de programação)/Distribuições de probabilidade