MATH266: Central Limit Theorem Demo

Simulating from Known Distributions

During past few weeks of class we have taken a tour of well known distributions, such as:

• Bernoulli and Binomial
• Continuous Uniform
• Gamma, Exponential, and Chi-squared
• Normal

We can use R, a statistical programming software/language to simulate draws from these distributions. R has many built in functions!

# Generating data from built in distributions: 
# Binomial: rbinom(n, size, prob)
# Uniform: runif(n, min, max)
# Chi-squared: rchisq(n, df)
# Normal: rnorm(n, mean, sd)

nsim<-10000
binom5<-rbinom(nsim, 10, .5)
unif5<-runif(nsim, 0, 10)
chisq5<-rchisq(nsim, 5)
norm5<-rnorm(nsim, 5, 2)

# Create a dataframe to store these data
dist_df<-data.frame(Data=c(binom5, unif5, chisq5, norm5), 
                    Distribution=c(rep("Binomial", nsim), 
                            rep("Uniform", nsim),
                            rep("ChiSquared", nsim),
                            rep("Normal", nsim)))

# We are using the tidyverse package for visualization with ggplot
library(tidyverse)

## Warning: package 'tidyverse' was built under R version 3.6.2

## Warning: package 'ggplot2' was built under R version 3.6.2

## Warning: package 'tibble' was built under R version 3.6.2

## Warning: package 'tidyr' was built under R version 3.6.2

## Warning: package 'readr' was built under R version 3.6.2

## Warning: package 'purrr' was built under R version 3.6.2

## Warning: package 'dplyr' was built under R version 3.6.2

## Warning: package 'forcats' was built under R version 3.6.2

ggplot(dist_df, aes(Data, after_stat(density), fill=Distribution))+
  geom_histogram()+
  facet_grid(Distribution~., scales = "free_y")

Central Limit Theorem Demo

\[\bar{X}\sim Normal(\mu , \sigma/\sqrt{n}), n\rightarrow \infty\]

In class we showed that, for a “sufficiently large” sample size the distribution of sample means is approximately normal with mean \(\mu\) and standard deviation \(\sigma / \sqrt{n}\), regardless of the underlying distribution.

binomCLT<-c()
unifCLT<-c()
chisqCLT<-c()
normCLT<-c()
index<-c()

# Loops for different sample sizes: n=1, 5, 10, and 30
for(i in c(1, 5, 10, 30)){
  for(j in 1:nsim){
    binomCLT<-c(binomCLT, mean(rbinom(i, 10, .5)))
    unifCLT<-c(unifCLT, mean(runif(i, 0, 10)))
    chisqCLT<-c(chisqCLT, mean(rchisq(i, 5)))
    normCLT<-c(normCLT, mean(rnorm(i, 5, 2)))
    index<-c(index, i)
  }
}

# Creates a dataframe to store simulation
clt_df<-data.frame(Data=c(binomCLT, unifCLT, chisqCLT, normCLT), 
                    Distribution=c(rep("Binomial", nsim*4), 
                                   rep("Uniform", nsim*4),
                                   rep("ChiSquared", nsim*4),
                                   rep("Normal", nsim*4)),
                   SampSize=rep(index, 4))

# Histograms
ggplot(clt_df, aes(Data, after_stat(density), fill=Distribution))+
  geom_histogram()+
  facet_grid(Distribution~SampSize, scales = "free_y")

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

# Approximate Density Curves
ggplot(clt_df, aes(Data, after_stat(density), fill=Distribution))+
  geom_density()+
  facet_grid(Distribution~SampSize, scales = "free_y")