**Question 1(ii) : Plot a histogram of the means of the samples generated in part (i), using anappropriate option in R for plotting the histogram on the probability densityscale.**

The task is to plot a histogram of the means of the samples generated in part (i) using R, and the histogram should be plotted on the probability density scale. Here’s the code and an explanation of the solution:

```
# Load the previously generated samples
load("exp_samples.RData")
# Calculate the means of each group of samples
sample_means <- tapply(samples, rep(1:num_samples, each = sample_size), mean)
# Plot the histogram on the probability density scale
hist(sample_means,
main = "Histogram of Sample Means (Exp(3) Distribution)",
xlab = "Sample Means",
ylab = "Probability Density",
prob = TRUE, # Set prob = TRUE for probability density scale
col = "skyblue", # Color of the bars
border = "black", # Color of the bar borders
breaks = 30) # Number of histogram bins
```

Explanation of the code and solution:

**Load the Previously Generated Samples**:`load("exp_samples.RData")`

loads the previously generated samples from the “exp_samples.RData” file. This step is necessary because we need the data for further analysis and visualization.**Calculate Sample Means**:`sample_means <- tapply(samples, rep(1:num_samples, each = sample_size), mean)`

calculates the means of each group of samples. The`tapply`

function is used to apply the`mean`

function to each group of samples. The`rep`

function is used to create a vector that repeats the sample numbers (from 1 to the number of samples) for each group. This gives us a vector of sample means.**Plot the Histogram**:`hist(sample_means, ...)`

plots the histogram of the sample means.`main`

,`xlab`

, and`ylab`

are used to add a title and labels to the plot.`prob = TRUE`

is the key option for plotting the histogram on the probability density scale. When`prob`

is set to`TRUE`

, the heights of the histogram bars are scaled so that the total area under the histogram equals 1, making it a probability density histogram.`col`

and`border`

specify the colors of the bars and their borders, respectively.`breaks`

specifies the number of histogram bins.

In summary, the code loads the previously generated sample data, calculates the sample means, and then plots a histogram of the sample means. The key aspect of this histogram is that it’s plotted on the probability density scale, meaning that the area under the histogram represents probabilities, making it suitable for visualizing the distribution of sample means from the Exponential distribution.

Check out the answer to previous question here

Stay Tuned for Rest of the answers from IFOA Actuary CS1B Exam.