# 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 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


### Bernouilli

We can draw from a Bernouilli 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 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

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


### Multinomial

> 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


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## Continuous distributions

### Beta and Dirichlet distributions

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


### Cauchy

We can sample n values from a Cauchy distribution with a given location parameter $x_0$ (default is 0) and scale parameter $\gamma$ (default is 1) using the rcauchy() function.

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


### Chi Square distribution

Quantile of the Chi square distribution ($\chi^2$ 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 $\theta$ using the rgamma() function. Alternatively a shape parameter and rate parameter $\beta=1/\theta$ can be given.

> x <- rgamma(n=100, 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 $\mu$ (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 $\mu$ (default is 1) using the rweibull() function.

> x <- rweibull(n=100, shape=0.5, scale=1)


### Extreme values and related distribution

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.

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

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## References

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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