UC Business Analytics R Programming Guide

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Generating Sequence of Random Numbers

Simulation is a common practice in data analysis. Sometimes your analysis requires the implementation of a statistical procedure that requires random number generation or sampling (i.e. Monte Carlo simulation, bootstrap sampling, etc). R comes with a set of pseudo-random number generators that allow you to simulate the most common probability distributions such as:


Uniform numbers

To generate random numbers from a uniform distribution you can use the runif() function. Alternatively, you can use sample() to take a random sample using with or without replacements.

# generate n random numbers between the default values of 0 and 1
runif(n)            

# generate n random numbers between 0 and 25
runif(n, min = 0, max = 25)       

# generate n random numbers between 0 and 25 (with replacement)
sample(0:25, n, replace = TRUE)   

# generate n random numbers between 0 and 25 (without replacement)
sample(0:25, n, replace = FALSE)

For example, to generate 25 random numbers between the values 0 and 10:

runif(25, min = 0, max = 10) 
##  [1] 9.2494720 1.0276421 9.6061007 7.4582455 8.3666868 0.8090925 7.5638221
##  [8] 4.2810155 2.5850736 9.7962788 6.1705894 0.7037997 9.5056240 4.7589622
## [15] 7.9750129 5.3932881 5.1624935 1.2704098 8.7064680 8.6649293 0.1049461
## [22] 1.4827342 2.7337917 7.5236131 3.9803653

For each non-uniform probability distribution there are four primary functions available to generate random numbers, density (aka probability mass function), cumulative density, and quantiles. The prefixes for these functions are:

  • r: random number generation
  • d: density or probability mass function
  • p: cumulative distribution
  • q: quantiles


Normal Distribution Numbers

The normal (or Gaussian) distribution is the most common and well know distribution. Within R, the normal distribution functions are written as `norm()`.

# generate n random numbers from a normal distribution with given mean & st. dev.
rnorm(n, mean = 0, sd = 1)    

# generate CDF probabilities for value(s) in vector q 
pnorm(q, mean = 0, sd = 1)    

# generate quantile for probabilities in vector p
qnorm(p, mean = 0, sd = 1)    

# generate density function probabilites for value(s) in vector x
dnorm(x, mean = 0, sd = 1)

For example, to generate 25 random numbers from a normal distribution with mean = 100 and standard deviation = 15:

x <- rnorm(25, mean = 100, sd = 15) 
x
##  [1] 107.84214 101.10742  73.67151 113.94035 108.47938  77.48445  73.02016
##  [8]  81.02323 101.64169 112.67715 105.28478  92.35393  85.96284 108.83169
## [15]  88.71057 115.13657 141.69830  99.91198 118.69664 110.61667  83.20282
## [22] 113.91008 109.10879  93.45276 109.01996

summary(x)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   73.02   88.71  105.30  101.10  110.60  141.70

You can also pass a vector of values. For instance, say you want to know the CDF probabilities for each value in the vector x created above:

pnorm(x, mean = 100, sd = 15) 
##  [1] 0.69944664 0.52942643 0.03960976 0.82364789 0.71406244 0.06667308
##  [7] 0.03603657 0.10291447 0.54357552 0.80098468 0.63770038 0.30511760
## [13] 0.17468526 0.72199534 0.22583658 0.84353778 0.99728111 0.49765904
## [19] 0.89369904 0.76045844 0.13139693 0.82312464 0.72815841 0.33124331
## [25] 0.72619004


Binomial Distribution Numbers

This is conventionally interpreted as the number of successes in size = x trials and with prob = p probability of success:

# generate a vector of length n displaying the number of successes from a trial 
# size = 100 with a probabilty of success = 0.5
rbinom(n, size = 100, prob = 0.5)  

# generate CDF probabilities for value(s) in vector q
pbinom(q, size = 100, prob = 0.5) 

# generate quantile for probabilities in vector p
qbinom(p, size = 100, prob = 0.5) 

# generate density function probabilites for value(s) in vector x
dbinom(x, size = 100, prob = 0.5)


Poisson Distribution Numbers

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occuring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

# generate a vector of length n displaying the random number of events occuring 
# when lambda (mean rate) equals 4.
rpois(n, lambda = 4)  

# generate CDF probabilities for value(s) in vector q when lambda (mean rate) 
# equals 4.
ppois(q, lambda = 4)  

# generate quantile for probabilities in vector p when lambda (mean rate) 
# equals 4.
qpois(p, lambda = 4)  

# generate density function probabilites for value(s) in vector x when lambda 
# (mean rate) equals 4.
dpois(x, lambda = 4)


Exponential Distribution Numbers

The Exponential probability distribution describes the time between events in a Poisson process.

# generate a vector of length n with rate = 1
rexp(n, rate = 1)   

# generate CDF probabilities for value(s) in vector q when rate = 4.
pexp(q, rate = 1)   

# generate quantile for probabilities in vector p when rate = 4.
qexp(p, rate = 1)   

# generate density function probabilites for value(s) in vector x when rate = 4.
dexp(x, rate = 1)


Gamma Distribution Numbers

The Gamma probability distribution is related to the Beta distribution and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

# generate a vector of length n with shape parameter = 1
rgamma(n, shape = 1)   

# generate CDF probabilities for value(s) in vector q when shape parameter = 1.
pgamma(q, shape = 1)   

# generate quantile for probabilities in vector p when shape parameter = 1.
qgamma(p, shape = 1)   

# generate density function probabilites for value(s) in vector x when shape 
# parameter = 1.
dgamma(x, shape = 1)