n=1000
pi1=0.5
population <- sample(1:2, n, replace = TRUE, prob = c(pi1, 1- pi1))
table(population)
n1 <- table(population)[1]; n2 <- table(population)[2]

mu.1 <- 2.5; sigma.1 <- 1
mu.2 <- 7; sigma.2 <- 2
x1 <- rnorm(n1, mu.1, sigma.1)
x2 <- rnorm(n2, mu.2, sigma.2)
x12 <- c(x1, x2)

mean(x1); mean(x2); mean(x12)
sd(x1); sd(x2); sd(x12)

par(mfrow=c(1,2))
hist(x1, freq = F); curve(dnorm(x, mean = mu.1, sd = sigma.1), add = T, lwd = 2, col = 'red')
hist(x2, freq = F); curve(dnorm(x, mean = mu.2, sd = sigma.2), add = T, lwd = 2, col = 'blue')
rug(x1, col = 'red')
rug(x2, col = 'blue')


hist(x12, freq = F,add=T)
curve(dnorm(x, mean = mu.1, sd = sigma.1),xlim=c(-3.5,10), ylim=c(0,0.5), lty = 1, col = 'red',ylab='densities', main='marginales')
curve(dnorm(x, mean = mu.2, sd = sigma.2), lty = 1,add=T, col = 'blue')


g.mixture <- function(x, pi, mu, sigma) {
  g <- pi * dnorm(x, mu[1], sigma[1]) +
    (1 - pi) * dnorm(x, mu[2], sigma[2])
  return(g)
}
curve(g.mixture(x, pi = pi1, c(mu.1, mu.2),
                c(sigma.1, sigma.2)), lwd = 2, add = T)

boundary=function(x)
{dnorm(x,mu.1,sigma.1)/dnorm(x,mu.2,sigma.2)-1
}


curve(boundary(x), lty = 1,add=T, col = 'black')


library(rootSolve) #required by the function uniroot.all
raices <- uniroot.all(boundary,c(-100,100))


boundary=function(x)
{dnorm(x,mu.1,sigma.1)/dnorm(x,mu.2,sigma.2)-2
}

library(rootSolve) #required by the function uniroot.all
raices <- uniroot.all(boundary,c(-1000,1000))


x.test =c(0, 1, 3.5, 8, 10, 12)
clasificacion <- ifelse(pi1 * dnorm(x.test, mu.1, sigma.1) >
                          (1 - pi1) * dnorm(x.test, mu.2, sigma.2), "Group 1", "Group 2")
cbind(x.test, poblacion = clasificacion)

################################################################
#Ejercicio: Calcular el error de clasificación sobre la muestra#
################################################################