4 Simulating random numbers

There are a number of functions in R that you can use to simulate random numbers according to different probability distributions.

The function sample allows you to take a sample of the specified size from the elements of a vector xusing sampling with or without replacement. You can use ?sample to read the documentation describing the command.

In the following, we will use the sample function to make 10,000 draws from the set of numbers 1, 2, 3 and 4 and display the distribution of the sampled values using a histogram.

First, we define a vector called x which contains the numbers 1, 2, 3, and 4. The function c allows us to combine these four numbers together into one vector:

x <- c(1, 2, 3, 4)

We now use the function sample to pick from those four numbers in x 10,000 times. The result, the 10,000 numbers chosen, is store in out:

out <- sample(x, 10000, replace=TRUE)

Lets plot a histogram of the values picked:

hist_out <- hist(out, main = '', xlab = 'Values', ylab = 'Frequency')

We picked each number with equal probability so the histogram shows each number is equally likely to have been chosen.

Problem

What is the difference in the output out1 and out2 in the following piece of code?

x <- c( 1, 2, 2, 3, 4, 1, 6, 7, 8, 10, 5, 5, 1, 4, 9 )
out1 <- sample(x, 10, replace=FALSE)
out2 <- sample(x, 10, replace=TRUE)

The option replace=TRUE activates sampling with replacement (i.e. the numbers that are picked are put back and can be picked again).

The option replace=FALSE activates sampling without replacement (i.e. the numbers that are picked are not put back and cannot be picked again).

Problem

Use the sample or sample.int function to simulate values from rolls of an unbiased six-sided die. Show that the distribution of values you obtain is consistent with an unbiased die.

Hint 1: Type ?sample.int in the console to get help on this function.

Hint 2: You may find it useful to use the function table. Type ?table in the console to get help on this function.

rolls_from_sample = sample(c(1:6), size=5000, replace=TRUE)
rolls_from_sample.int = sample(6, size=5000, replace=TRUE)

table(rolls_from_sample)

## rolls_from_sample
##   1   2   3   4   5   6 
## 847 845 841 836 814 817

table(rolls_from_sample.int)

## rolls_from_sample.int
##   1   2   3   4   5   6 
## 852 810 880 837 820 801

Both gives a uniform distribution over the numbers 1-6. The function sample.int is a specialised version of sample for sampling integers. Many R libraries have specialised versions of more general functions to do specific tasks under certain conditions.