# 8 Random Numbers

In this lesson we learn about generating “random” numbers in R.

## 8.1 Generating “Random” Numbers

The random word is between quotes above because R - or any software - cannot really generate truly random numbers. It gets really close though. This is because softwares use an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. This is termed a pseudorandom number generator, which is not truly random, because it is completely determined by a relatively small set of initial values, called the seed.

### 8.1.1 Sampling from a vector

Below we will sample from several vectors, and emulate tossing a coin, rolling a die, playing the lottery, etc.

[1] 0
 [1] 0 0 1 1 0 0 1 0 1 0
[1] 2
 [1] 1 6 1 6 1 1 1 4 2 1
 [1]  7 15  1 12  6  5  3  7  4  3
[1] 22  2 34  5 43  6
[1] 12 49 50 15  7 40
[1] "King of Spades"

### 8.1.2 Concept: Randomness & Distributions

In computational statistics, and in R, random numbers are described by a distributions, which is a function specifying the probability that a random number is within some range. If the random number is continuous this is called “probability density function”, if the random number is discrete then the term is “probability mass function”.

If you want to learn more about Probability density functions, you can run the code below in your RStudio, which will produce a Shiny App displaying the probability density functions for the Normal, Poisson and Beta distributions. Here’s the link if you want the raw App

### 8.1.3 Sampling from a Distribution

How to choose a random number in R? As a language for statistical analysis, R has a comprehensive library of functions for generating random numbers from various statistical distributions.

Distribution R Function
Uniform runif
Normal rnorm
Student’s t rt
F rf
chi-squared rchisq
Exponential rexp
Log normal rlnorm
Beta rbeta
Binomial rbinom
Negative Binomial rnbinom
Poisson rpois
Gamma rgamma
Weibull rweibull
Cauchy rcauchy
Multinomial rmultinom
Geometric rgeom
?Distributions full list

The Uniform Distribution

If you want to generate a decimal number where any value (including fractional values) between the stated minimum and maximum is equally likely, the runif() function is what you are looking for.This function generates values from the Uniform distribution. Here’s how to generate one random number between 0 and 1:

[1] 0.9159742

Of course, when you run this, you’ll get a different number, but it will definitely be between 0 and 1. You won’t get the values 0 or 1 exactly, either.

If you want to generate multiple random values, you can generate several values at once by specifying the number of values you want as the first argument to runif. Here’s how to generate 10 values between 0 and 1.

 [1] 0.9945982 0.9423607 0.4861354 0.2834595 0.2515457 0.5032552 0.4969662
[8] 0.3184458 0.9622228 0.6340994

The Normal Distribution

$\varphi \left(z\right)=\frac{1}{\sqrt{2\phantom{\rule{thinmathspace}{0ex}}\pi }}{e}^{-\frac{1}{2}{z}^{2}},\phantom{\rule{1em}{0ex}}z\in \mathbb{R}$

To generate numbers from a normal distribution, use rnorm().

[1] -1.138608
[1]  1.367827179  1.329564791  0.336472797  0.006892838 -0.455468738
[1] 0.08299672
[1] 4.667343
[1] -0.4667063  9.2337872  2.4872527  1.6287116 14.1312977
 [1]  12.3502131   9.6674139 -17.3221952   9.0020941  19.7603173
[6]  14.1386892  19.1232216  29.8373220  21.6910851   4.9126298
[11]  17.0418018   8.0158373   4.6192921 -18.5575866   2.1035315
[16]  14.8781464  31.6803254  15.0069461  16.2021020   0.3409679
[21]  11.6265471 -10.7823754  14.8522682  16.9676878  11.8551392
[26]  17.0073352  13.1168103  17.6046236  28.4246363  21.1236284
[31]  10.3266396  -1.1444896  14.1805782   5.9976476  24.9349310
[36]  -6.0708094   5.8424821  14.2200837   8.4826346   3.9384888
[41]   6.9527893  16.2953610  18.9517198  16.6021263  32.7348352
[46]  21.7349757  12.8770973   3.4022991  39.1914013  16.7741550
[51]   3.1567966  11.8649208   6.7560670   7.2529578   0.6649666
[56]  11.1684534  13.1916024  -0.7754212 -22.3315213   7.4512535
[61]  10.2951783  15.9427377  10.5913517  14.1339889  -0.9777217
[66]  17.1117526  17.1888873  12.5165107  23.5727444  14.0446847
[71]  12.6436427  12.6804390  14.3693058  20.6012391  14.5219040
[ reached getOption("max.print") -- omitted 25 entries ]

The Poisson Distribution

To generate numbers from a poisson distribution, use rpois(). The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space. The Poisson distribution may be useful to model events such as:

• The number of meteors greater than 1 meter diameter that strike earth in a year
• The number of occurrences of the DNA sequence “ACGT” in a gene
• The number of patients arriving in an emergency room between 11 and 12 pm

In probability theory, a Poisson process is a stochastic process that counts the number of independent events in a given time interval. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

 [1] 1 2 1 1 0 1 0 1 0 1
 [1] 7 4 6 3 6 6 6 1 4 4
 [1] 0 0 1 0 3 0 0 0 0 1
 [1] 7 1 6 3 7 3 6 5 6 3

## 8.2 13.1 Wrap-up Exercise

Let’s say I have a categorical variable which can take the values A, B, C and D. How can I generate 10000 random data points and control for the frequency of each? For example: A = 10% B = 20% C = 65% D = 5%. Any ideas how to do this? Don’t fret if you have no ideas. I didn’t when I first tried to solve it. But it helped me a great deal to practice the skills I learned and how they can be useful.

### 8.2.1 Solution 1: Elegant and quickest

x
A    B    C    D
1014 2073 6427  486 
x
A      B      C      D
0.1014 0.2073 0.6427 0.0486 

### 8.2.2 Solution 2: Clever, but dirty

 [1] "C" "C" "C" "C" "B" "C" "B" "C" "C" "C" "B" "C" "C" "C" "A" "C" "C"
[18] "C" "C" "C" "D" "C" "C" "C" "C" "B" "D" "D" "C" "C" "B" "B" "C" "C"
[35] "B" "B" "B" "B" "A" "B" "C" "C" "C" "A" "C" "B" "B" "C" "C" "C" "C"
[52] "D" "C" "B" "C" "C" "C" "C" "C" "C" "B" "A" "C" "B" "B" "C" "D" "C"
[69] "C" "C" "C" "C" "C" "B" "C"
[ reached getOption("max.print") -- omitted 925 entries ]
 [1] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
[18] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
[35] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
[52] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
[69] "A" "A" "A" "A" "A" "A" "A"
[ reached getOption("max.print") -- omitted 9925 entries ]
x
A    B    C    D
0.10 0.20 0.65 0.05 

### 8.2.3 Solution 3: Brute force (reversed thinking?)

x
A      B      C      D
0.0967 0.1901 0.6653 0.0479