simulation_central_limit_theorem.R

# ---------- Central Limit Theorem ----------
# Parameters
sample_mean = 100000
sample_size = 20
set.seed(1234)

# 1 Exponential Distribution
x = rexp(n = 4000, rate = 0.1)
hist(x)
mean(x)
sd(x)
dist = replicate(n = sample_mean,
                 expr = mean(sample(x = x,
                                    size = sample_size,
                                    replace = TRUE)))
hist(dist)
mean(dist)
sd(dist)
sd(x) / sqrt(sample_size) # Proof

# 2 Uniform Distribution
x = runif(n = 4000, min = 0, max = 1)
hist(x)
mean(x)
sd(x)
dist = replicate(n = sample_mean,
                 expr = mean(sample(x = x,
                                    size = sample_size,
                                    replace = TRUE)))
hist(dist)
mean(dist)
sd(dist)
sd(x) / sqrt(sample_size) # Proof

# 3 Normal Distribution
x = rnorm(n = 4000, mean = 0, sd = 1)
hist(x)
mean(x)
sd(x)
dist = replicate(n = sample_mean,
                 expr = mean(sample(x = x,
                                    size = sample_size,
                                    replace = TRUE)))
hist(dist)
mean(dist)
sd(dist)
sd(x) / sqrt(sample_size) # Proof

# 4 Binomial Distribution
x = rbinom(n = 4000, size = 500, prob = 0.7)
hist(x)
mean(x)
sd(x)
dist = replicate(n = sample_mean,
                 expr = mean(sample(x = x,
                                    size = sample_size,
                                    replace = TRUE)))
hist(dist)
mean(dist)
sd(dist)
sd(x) / sqrt(sample_size) # Proof

# 5 Chisquare Distribution
x = rchisq(n = 4000, df = 10, ncp = 1)
hist(x)
mean(x)
sd(x)
dist = replicate(n = sample_mean,
                 expr = mean(sample(x = x,
                                    size = sample_size,
                                    replace = TRUE)))
hist(dist)
mean(dist)
sd(dist)
sd(x) / sqrt(sample_size) # Proof