R Programming/Probability Distributions

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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[edit | edit source]

Benford Distribution[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

We can sample n times from a hypergeometric distribution using the rhyper() function.

> x <- rhyper(n=1000, 15, 5, 5)

Geometric distribution[edit | edit source]

The geometric distribution.

> N <- 10000
> x <- rgeom(N, .5)
> x <- rgeom(N, .01)

Multinomial[edit | edit source]

The multinomial distribution.

> sample(1:6, 100, replace=T, prob= rep(1/6,6))

Negative binomial distribution[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

Beta and Dirichlet distributions[edit | edit source]

>library(gtools)
>?rdirichlet
>library(bayesm)
>?rdirichlet
>library(MCMCpack)
>?Dirichlet

Cauchy[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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[edit | edit source]

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 | edit source]

  • 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 | edit source]

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 | edit source]

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 | edit source]

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[edit | edit source]

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 | edit source]

  • Functions for circular statistics are included in the CircStats package.
    • dvm() Von Mises (also known as the nircular normal or Tikhonov distribution) density function
    • dtri() triangular density function
    • dmixedvm() Mixed Von Mises density
    • dwrpcauchy() wrapped Cauchy density
    • dwrpnorm() wrapped normal density.

See also[edit | edit source]

  • Packages VGAM, SuppDists, actuar, fBasics, bayesm, MCMCpack

References[edit | edit source]

  1. Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, 78, 551–572.
  2. Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4, 39–40.
  3. 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.
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