Benford Distribution

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

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

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 size^[3]. dzipf() and pzipf() (VGAM)

> library(VGAM)
> dzipf(x=2, N=1000, s=2)

Beta and Dirichlet distributions

• Beta 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 ${\displaystyle x_{0}}$ (default is 0) and scale parameter ${\displaystyle \gamma }$ (default is 1) using the rcauchy() function.

> x <- rcauchy(n=100, location=0, scale=1)

Chi Square distribution

Quantile of the Chi-square distribution (${\displaystyle \chi ^{2}}$ 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 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 ${\displaystyle \theta }$ using the rgamma() function. Alternatively a shape parameter and rate parameter ${\displaystyle \beta =1/\theta }$ 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 ${\displaystyle \mu }$ (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 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

• 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 (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 ${\displaystyle \mu }$ (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 (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.