STAT 5104 HW1

Name : Mst Nigar Sultana

n = 5        # Number of coin flips
p = 0.5      # Probability of heads
i_vals = 0:5 # Possible values for number of heads (0 to 5)

theoretical_prob = dbinom(i_vals, size = n, prob = p)

set.seed(123)  # Set seed for reproducibility
n_trials = 1000  # Number of trials

results = replicate(n_trials, sum(sample(c(0, 1), n, replace = TRUE)))

outcomes <- table(factor(results, levels = 0:5))

# Empirical probabilities (frequency / total trials)
empirical_prob = outcomes / n_trials

# Comparison between theoretical and empirical probabilities
comparison = data.frame(
  Heads = i_vals,
  Theoretical = theoretical_prob,
  Empirical = as.numeric(empirical_prob)
)

print(comparison)

library(ggplot2)

comparison = data.frame(
  Heads = factor(i_vals),  # Treat Heads as a categorical variable
  Theoretical = theoretical_prob,
  Empirical = as.numeric(empirical_prob)
)
# Reshape data for easy plotting with ggplot
comparison_melted <- reshape2::melt(comparison, id.vars = "Heads")

ggplot(comparison_melted, aes(x = Heads, y = value, fill = variable)) +
  geom_bar(stat = "identity", position = "dodge") +
  labs(title = "Comparison of Theoretical and Empirical Probabilities",
       x = "Number of Heads",
       y = "Probability") +
  scale_fill_manual(values = c("Theoretical" = "#9999CC", "Empirical" = "#66CC99"),
                    name = "Type",
                    labels = c("Theoretical", "Empirical")) +
  theme_minimal()