Game 11: Central Limit Theorem

An Illustration of CLT

We adapt the code below and modify it to show normal probablity plot and add some mixture/monstrous distributions

Code source:

• Sampling Distributions and Central Limit Theorem in R

sdm.sim <- function(n,src.dist=NULL,param1=NULL,param2=NULL) {
   r <- 10000  # Number of replications/samples - DO NOT ADJUST
   # This produces a matrix of observations with  
   # n columns and r rows. Each row is one sample:
   my.samples <- switch(src.dist,
    "E" = matrix(rexp(n*r,param1),r),
    "N" = matrix(rnorm(n*r,param1,param2),r),
    "U" = matrix(runif(n*r,param1,param2),r),
    "P" = matrix(rpois(n*r,param1),r),
    "C" = matrix(rcauchy(n*r,param1,param2),r),
  "B" = matrix(rbinom(n*r,param1,param2),r),
    "G" = matrix(rgamma(n*r,param1,param2),r),
    "X" = matrix(rchisq(n*r,param1),r),
    "T" = matrix(rt(n*r,param1),r),
    "Z" = matrix((rnorm(n*r,param1,param2))^2,r),
    "ZZ"=matrix(rnorm(n*r,param1,param2)+5*runif(n*r,param1,param2),r)
    ## ZZ is our experimental construction by mixing normal and uniform together
    )
   
   all.sample.sums <- apply(my.samples,1,sum)
   all.sample.means <- apply(my.samples,1,mean)   
   all.sample.vars <- apply(my.samples,1,var) 
   par(mfrow=c(2,2))
   hist(my.samples[1,],col="gray",main="Distribution of One Sample")
   hist(all.sample.sums,col="gray",main="Sampling Distributionof
    the Sum")
   hist(all.sample.means,col="gray",main="Sampling Distributionof the Mean")
   qqnorm(all.sample.means, main="NormalProbabilityPlot of the Mean")
#   hist(all.sample.vars,col="gray",main="Sampling Distributionnof
#   the Variance")
}

Select source distribution as exponential

sdm.sim(2,src.dist="E",param1=1)

curve(dexp(x,1),0,3)

Uniform

sdm.sim(500,src.dist="U",param1=0,param2=1)

Bernoulli and Binomial

sdm.sim(2,src.dist="B",param1=1,param2=.4)

sdm.sim(200,src.dist="B",param1=1,param2=.5)

Normal

sdm.sim(2,src.dist="N",param1=1,param2=4)

sdm.sim(200,src.dist="Z",param1=0,param2=1)

ZZ

sdm.sim(200,src.dist="ZZ",param1=0,param2=1)