# Review Chapter 4

# Normal density Section 4.1.1

xvar = seq(from=-3,to=9,length=1000)

norm1 = (1/sqrt(2*pi))*exp(-(1/2)*(xvar-1)**2)
n1 = (1/sqrt(2*pi))*exp(-(1/2)*(2.64-1)**2)
norm2 = (1/sqrt(2*pi))*exp(-(1/2)*(xvar-4)**2)
n2 = (1/sqrt(2*pi))*exp(-(1/2)*(2.64-4)**2)

plot(xvar, norm1, type="l", lwd=2,   main="",xlab="x-axis",ylab="densities",  col="blue")
lines(xvar,norm2,type="l",lwd=2,col="red")



lines(c(2.64,2.64),c(0,n1),col="blue",lwd=2)

lines(c(2.64,2.64),c(n1,n2),col="red",lwd=2)

#############################################

# The likelihood ratio test
# Section 4.2.1 
# Normal distribution (using data from above)

LR = norm2/norm1

plot(xvar, log(LR), type="l", lwd=2,   main="Log likelihood ratio",xlab="x-axis",ylab="log ratio",  col="blue")

lines(c(2.64,2.64),c(-15,20),col="red",lwd=2,lty=2)
lines(c(-2,8),c(0.42,0.42),col="red",lwd=2,lty=2)

#lines(xvar, norm1, type="l", lwd=2,col="blue")
#lines(xvar,norm2,type="l",lwd=2,col="red")

##########################################

# Binomial distribution
# Section 4.2.2

N2=dbinom(c(0:10),10,0.8)
N1=dbinom(c(0:10),10,0.6)
N0=dbinom(c(0:10),10,0.3)

N0sum = (1-pbinom(c(0:10),10,0.3))+dbinom(c(0:10),10,0.3)
N1sum = (1-pbinom(c(0:10),10,0.6))+dbinom(c(0:10),10,0.6)
N2sum = (1-pbinom(c(0:10),10,0.8))+dbinom(c(0:10),10,0.8)

round(N2,4)

round(log(N2/N0),4)

plot(c(0:10),N0, type="o", lwd=2,   main="Binomial",xlab="outcome",ylab="probability",  col="blue")
points(c(0:10),N1, type="o", lwd=2, col="red")
lines(c(0:10),N1, type="o", lwd=2, col="red")

n1 =dbinom(6,10,0.6)
n0 =dbinom(6,10,0.3)

lines(c(6,6),c(0,n0),col="blue",lwd=2)
lines(c(6,6),c(n0,n1),col="red",lwd=2)

###

LR = N1/N0

plot(c(0:10), log(LR), type="o", lwd=2,   main="Log likelihood ratio",xlab="x-axis",ylab="log ratio",  col="blue")
lines(c(6,6),c(-6,log(n1/n0)),col="blue",lwd=2)

###########################################################

# None-monotonic functions
# Section 4.2.4


xvar = seq(from=-3,to=9,length=1000)

norm0 = (1/sqrt(2*pi))*exp(-(1/2)*(xvar-2.5)**2)
norm1 = (1/sqrt(2*pi))*exp(-(1/2)*(xvar-1)**2)
n1 = (1/sqrt(2*pi))*exp(-(1/2)*(2.64-1)**2)
norm2 = (1/sqrt(2*pi))*exp(-(1/2)*(xvar-4)**2)
n2 = (1/sqrt(2*pi))*exp(-(1/2)*(2.64-4)**2)
norm3 = 0.5*norm1 + 0.5*norm2


plot(xvar, norm0, type="l", lwd=2,   main="",xlab="x-axis",ylab="densities",  col="blue")
lines(xvar,norm3,type="l",lwd=2,col="red")

plot(xvar, log(norm3/norm0), type="l", lwd=2,   main="",xlab="x-axis",ylab="log likelihood ratio",  col="blue")