#==============================================================================
# Comparacoes da distribuicao t-Student para diferentes graus de liberade e a 
# distribuicao normal padrao
#==============================================================================
xseq   <- seq(-5,5,length=10001)
zseq   <- dnorm(xseq)
yseq1  <- dt(xseq,1)
yseq5  <- dt(xseq,5)
yseq15 <- dt(xseq,15)

par(mfrow=c(1,1),lwd=2.0,cex.lab=1.5,cex.axis=1.5,lab=c(10,5,5),
    mar=c(5,5,2,2.5),xpd=T,cex.main=2.0)
plot(xseq,zseq,type="l",lwd=4,col="black",xlab="x",
  ylab=expression(f(x)),ylim=c(0,0.42))
lines(xseq,yseq1, lwd=4,col="red")
lines(xseq,yseq5, lwd=4,col="blue")
lines(xseq,yseq15,lwd=4,col="magenta")
legend("topleft",lty=rep(1,4),lwd=rep(4,4),bty="n",
  legend=c("N(0; 1)",expression(t[1]),expression(t[5]),expression(t[15])),
  col=c("black","red","blue","magenta"))

#==============================================================================
# Sequencia de valores gerados da t-Student t(0; 1; df) para diferentes graus 
# de liberdade (df) (verificando a aproximacao para normal padrao)
#==============================================================================
set.seed(5438)
n    <- 10000      #tamanho da amostra
x1   <- rt(n,df=1)
x2   <- rt(n,df=2)
x3   <- rt(n,df=3)
x5   <- rt(n,df=5)
x15  <- rt(n,df=15)
x30  <- rt(n,df=30)

ts.plot(x1, ylim=c(-90000,90000),xlab="Índice",ylab=expression(x[1]))
hist(x1,prob=T,nclass=30,main="",xlab=expression(x[1]),ylab="Densidade")

ts.plot(x2, ylim=c(-200,200),xlab="Índice",ylab=expression(x[2]))
hist(x2,prob=T,nclass=30,main="",xlab=expression(x[2]),ylab="Densidade")

ts.plot(x3, ylim=c(-60,60),xlab="Índice",ylab=expression(x[3]))
hist(x3,prob=T,nclass=30,main="",xlab=expression(x[3]),ylab="Densidade")

ts.plot(x5, ylim=c(-20,20),xlab="Índice",ylab=expression(x[5]))
hist(x5,prob=T,nclass=30,main="",xlab=expression(x[5]),ylab="Densidade")

ts.plot(x15, ylim=c(-6,6),xlab="Índice",ylab=expression(x[15]))
hist(x15,prob=T,nclass=30,main="",xlab=expression(x[15]),ylab="Densidade")

ts.plot(x30, ylim=c(-6,6),xlab="Índice",ylab=expression(x[30]))
hist(x30,prob=T,nclass=30,main="",xlab=expression(x[30]),ylab="Densidade")

boxplot(x15,main=expression(t[15]),ylab=expression(x[15]),pch=15)
boxplot(x30,main=expression(t[30]),ylab=expression(x[30]),pch=15)
qqnorm(x15,xlab="Percentis teóricos",ylab="Percentis amostrais",
  main=expression(t[15]),pch=15)
qqline(x15,lwd=2,col="red")
qqnorm(x30,xlab="Percentis teóricos",ylab="Percentis amostrais",
  main=expression(t[30]),pch=15)
qqline(x30,lwd=2,col="red")