Distributions

Statistics frequently utilize the concept of distributions that come from probability theory. These distribution represent infinite, continuous populations, that we can sample from.

Distributions are organized into families, which is a set of rules that describe the expected frequency of values. A specific distribution is given by a distribution family and its parameters.

The uniform distributions

For instance, one of the most frequently used distribution families is the uniform distribution family, which can be described with two parameters: a minimum (a) and a maximum (b). The uniform distribution expresses that if you would draw random numbers from the continuous interval between a and b, every number - with the minimum and the maximum included - has exactly equal chance of getting drawn.

The simplest, and most important distribution in the uniform distribution family is the standard uniform distribution: U(0,1) - U represent the the distribution family, 0 and 1 are the parameters.

Note: Almost all implementations of (pseudo)random number generation are actually derived from this simple distribution!

Sampling from the uniform distribution

We can sample from this distribution in R using the runif() function. The first argument of this function (x) sets the number of elements that we want to draw from this distribution.

# getting 5 values from U(0,1)
runif(5)

[1] 0.0001456398 0.9648580905 0.3221379726 0.3516660845 0.3109391942

Note that the results of this random draw will not be ordered, and every time you run the function, you will get a different result.

Other uniform distributions

You can use the usual question mark ?runif command to acces the manual file fo this function. If you look up the usage of the runif(), you will see that it actually allows you to set the minimum and maximum parameters, so you can generate random numbers from any other uniform distribution, by setting the min and max arguments.

# getting 5 values from U(10,20)
runif(3, min=10, max=20)

[1] 19.87434 12.28877 17.62724

Note that this manual page is shared with other functions dunif(), punif(), qunif() that generate other descriptors of these distributions. You can guess that the ‘-unif’ postfix indicates the uniform family. The r- prefix indicates random number generation.

Confirming results: large numbers

When only small samples are generated, they will be considerably uneven. However, you can quickly confirm the uniformity of this distributionif you create 1) a large enough sample and 2) make a histogram from it.

# getting one million values
bigSamp <- runif(1e6, min=10, max=20)
hist(bigSamp)

Other families

You can guess now that random number generation (and also the other distribution-functions) can be executed with other distributions as well!

These include, for example, the following frequently used distribution families (both discrete and continuous):

Family	Random number generation
Uniform	runif()
Binomial	rbinom()
Normal	rnorm()
Geometric	rgeom()
Exponential	rexp()
Lognormal	rlnorm()
Beta	rbeta()

See ?Distributions for a complete list.

Related exercises: