#Homework 3 - Problem Number 1#
#This code modifies newrt to count the number of rejections #
#  required before it actually produces the random number. #
#  It then counts the number of times required for each of #
#  2, 8, and 30 degrees of freedom.  It produces summary #           
#  statistics for the number of rejections and then produces #
#  a q-q plot for each of these.  It is not surprising that #
#  the normality assumption for either the t-test or ANOVA is #
#  not met.  One nonparametric test that could be used instead #
#  of a one-way ANOVA is the Kruskal-Wallis test. #
#  The inside front cover of Conover's nonparametric book has #
#  a nice list of when different tests should be used. #

newrt<-function(df=1){ 
#generates a t random variable using the accept-reject method#
   check<-0
   nrejects<--1
   while (check==0){
     nrejects<-nrejects+1
     y1<-runif(1)*2-1
     y2<-runif(1)
     if (y2<0.5){x<-y1}
     else {x<-1/y1}
     u<-runif(1)
     if (u<= (((1+(x^2)/df)^(-(df+1)/2) )/min(1,x^(-2))))
       {check<-1}
   }
   return(x,nrejects)
}

rejects2<-rep(0,500)
rejects8<-rep(0,500)
rejects30<-rep(0,500)
for (i in 1:500){
  rejects2[i]<-newrt(df=2)$nrejects
  rejects8[i]<-newrt(df=8)$nrejects
  rejects30[i]<-newrt(df=30)$nrejects
}
par(mfrow=c(3,1))
summary(rejects2)
summary(rejects8)
summary(rejects30)
qqnorm(rejects2)
qqnorm(rejects8)
qqnorm(rejects30)
kruskal.test(c(rejects2,rejects8,rejects30),c(rep(2,500),rep(8,500),rep(30,500)))




#Homework 3 - Problem Number 2#
#This code modifies mvrnorm to generate multivariate random #
#  variables using the correlation matrix and variance vector #
#  instead of the covariance matrix.  It then tries a brief #
#  example to make sure it works. #

library(MASS)

mvrnorm2<-function(n,mu,vars,corr){
#put some comments here#
  sigma<-corr*(sqrt(vars)%*%t(sqrt(vars)))
  mvrnorm(n,mu,sigma)}

mu<-c(5,84,13)
vars<-c(18,1,4)
corr<-matrix(c(1,.4,.7,.4,1,.9,.7,.9,1),nrow=3)
x<-mvrnorm2(5000,mu,vars,corr)
apply(x,2,mean)
var(x)
cor(x)