Exercise 5, Page 362

To achieve the task described, we’ll write an R program that performs the following steps:

  1. Chooses 25 random numbers from the range \([0, 20]\) (inclusive) 1000 times.

  2. Computes the sum \(S_{25}\) for each set of 25 numbers.

  3. Repeats the experiment 1000 times to generate a distribution of \(S_{25}\).

  4. Plots the density of \(S_{25}\) using a bar graph.

This approach allows us to explore the distribution of the sum of 25 randomly chosen numbers from a specified range, through repeated sampling and visualization techniques.

set.seed(123)

# Number of experiments
n_experiments <- 1000

# Generating the sums S25
S25_sums <- replicate(n_experiments, sum(sample(0:20, 25, replace = TRUE)))

# Plotting the density of S25 using a histogram to represent it as a bar graph
hist(S25_sums, breaks = 50, probability = TRUE, col = "blue",
     main = "Density of S25",
     xlab = "Sum S25",
     ylab = "Density")

lines(density(S25_sums), col = "red", lwd = 2)