R Notebook Writing Assignment Unit 2 Markdown Document

Instructions

Consider a set of Random numbers from the standard normal distribution in a vector-like X.

Step 1

Plot a histogram of 1000 random numbers from the standard normal distribution. (Keep breaks=50 and the Z Score=1.669)

# Load necessary libraries
library(ggplot2)

# Set the seed for reproducibility
set.seed(143)



# Create a data frame of Random Numbers that form a normal distribution


# Generate 1000 random numbers from the standard normal distribution
data <- data.frame(values = rnorm(1000))

Generate a Histogram of Random Numbers

# Plot a histogram of 1000 random numbers from the standard normal distribution. (Keep breaks=50 and the Z Score=1.669)

histogram <- ggplot(data, aes(x = values)) +
  geom_histogram(aes(fill = ifelse(values > -1.669 & values < 1.669, "red", "blue")), brakes = 50, 
                 col = "black", position = "identity") +
  scale_fill_manual(values = c("blue", "red")) +  # Assign colors to the specified ranges
  labs(title = "Histogram of Standard Normal Distribution", x = "Z Score", y = "Frequency")

# Print the histogram
print(histogram)

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Step 2

Now take 1000 samples of size 20 from the standard normal distribution at the same time. Use a matrix, where each row of X is a sample of size 20 from the standard normal distribution with 1000 rows.

# Generate 1000 samples of size 20 from the standard normal distribution
X <- matrix(rnorm(1000*20), ncol = 20, nrow = 1000)

histogram <- ggplot(data, aes(x = values)) +
  geom_histogram(aes(fill = ifelse(values > -1.669 & values < 1.669, "red", "blue")), brakes = 50,
                 col = "black", position = "identity") +
  scale_fill_manual(values = c("blue", "red")) +  # Assign colors to the specified ranges
  labs(title = "Histogram of Standard Normal Distribution", x = "Z Score", y = "Frequency")

# Print the histogram
print(histogram)

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Step 3

Repeat Step 1 with random numbers from an exponential distribution. (Take lambda=0.6)

# Set lambda value
lambda <- 0.6

# Number of random numbers to generate
num_samples <- 1000

# Generate random numbers from exponential distribution
data <- data.frame(values = rexp(num_samples, rate = lambda))

Draw the historam

# Create a histogram with 50 brakes
histogram <- ggplot(data, aes(x = values)) +
  geom_histogram(aes(fill = ifelse(values > -1.669 & values < 1.669, "red", "blue")), brakes = 50, bins = 30, 
                 col = "black", position = "identity") +
  scale_fill_manual(values = c("blue", "red")) +
  labs(title = "Histogram of Exponential Distribution", x = "Values", y = "Frequency")


# Print the histogram
print(histogram)

Step 4

Repeat the steps in Step 2 with random numbers from an exponential distribution. (Take lambda=0.6)

# Generate random numbers from exponential distribution
data <- data.frame(values = rexp(num_samples*20, rate = lambda))

# Create a histogram with 50 bins
histogram <- ggplot(data, aes(x = values)) +
  geom_histogram(aes(fill = ifelse(values > -1.669 & values < 1.669, "red", "blue")), brakes = 50,
                 col = "black", position = "identity") +
  scale_fill_manual(values = c("blue", "red")) +
  labs(title = "Histogram of Exponential Distribution", x = "Values", y = "Frequency")

Warning in geom_histogram(aes(fill = ifelse(values > -1.669 & values < 1.669, :
Ignoring unknown parameters: `brakes`

# Print the histogram
print(histogram)

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.