Visualizing the Normal approximation to the Binomial

norm_approx_binom

library(tidyverse)

df <- data.frame(x = 0:20, y = dbinom(0:20, size = 20, prob = .4))

# Plot pdf of Bin(20, 0.4)
ggplot(df, aes(x, y)) +
  geom_bar(stat = "identity", alpha = .4) + 
  geom_bar(stat = "identity", alpha = .9, data = subset(df, x > 4 & x <= 7)) +
  ylab("Binomial probability") +
  theme_minimal()

# Calculate binomial probability
pbinom(7, 20, .4) - pbinom(4, 20, .4)
  
# Normal approximation to Binomial
ggplot(df, aes(x, y)) +
  stat_function(fun = dnorm, args = list(mean = 20*.4, sd = sqrt(20*.4*.6))) +
  geom_bar(stat = "identity", alpha = .4) + ylab("Probability") +
  stat_function(fun = dnorm, args = list(mean = 20*.4, sd = sqrt(20*.4*.6)),
                xlim = c(4.5, 7.5), geom = "area", alpha = .6, fill = "blue") +
  theme_minimal() + ggtitle("Normal approximation (blue area)")

# Calculate normal probability
pnorm(7.5, mean = 20*.4, sd = sqrt(20*.4*.6)) - 
  pnorm(4.5, mean = 20*.4, sd = sqrt(20*.4*.6))

# Visualize both
ggplot(df, aes(x, y)) +
  stat_function(fun = dnorm, args = list(mean = 20*.4, sd = sqrt(20*.4*.6))) +
  geom_bar(stat = "identity", alpha = .4) + ylab("Probability") +
  geom_bar(stat = "identity", fill = "red", alpha = .9, data = subset(df, x > 4 & x <= 7)) +
  stat_function(fun = dnorm, args = list(mean = 20*.4, sd = sqrt(20*.4*.6)),
                xlim = c(4.5, 7.5), geom = "area", alpha = .6, fill = "blue") +
  theme_minimal() + ggtitle("Normal approximation (blue area) to Binomial (red area)")