set.seed(34567)
x <- rnorm(n = 50)
eps <- rnorm(n = 50)
# Responses
yLin <- -0.5 + 1.5 * x + eps
yQua <- -0.5 + 1.5 * x^2 + eps
yExp <- -0.5 + 1.5 * 2^x + eps
# Data
leastSquares <- data.frame(x = x, yLin = yLin, yQua = yQua, yExp = yExp)
# Plots
plot(yLin ~ x)
beta1_hat1 <- cov(x,yLin)/var(x)
beta0_hat1 <- mean(yLin) - beta1_hat1*mean(x)
beta1_hat1
beta0_hat1
#vamos a hallar la prediccion model1 y tambien hallamos el error
y_pred1 = beta0_hat1+beta1_hat1*(x)
sigma_hat1= sum((yLin-y_pred1)^2)/(length(x)-2)
model1<-lm(yLin ~ x, data = leastSquares)
summary(model1)
abline(model1)

# MODELO 2
plot(yQua ~ x)
beta1_hat2 <- cov(x,yQua)/var(x)
beta0_hat2 <- mean(yQua) - beta1_hat2*mean(x)
beta1_hat2
beta0_hat2
#vamos a hallar la prediccion model1 y tambien hallamos el error
y_pred2 = beta0_hat2+beta1_hat2*(x)
sigma_hat2= sum((yQua-y_pred2)^2)/(length(x)-2)
model2<-lm(yQua ~ x, data = leastSquares)
summary(model2)
abline(model2)

# MODELO 3
plot(yExp ~ x)
beta1_hat3 <- cov(x,yExp)/var(x)
beta0_hat3 <- mean(yExp) - beta1_hat3*mean(x)
beta1_hat3
beta0_hat3
#vamos a hallar la prediccion model1 y tambien hallamos el error
y_pred3 = beta0_hat3+beta1_hat3*(x)
sigma_hat3= sum((yExp-y_pred3)^2)/(length(x)-2)
model3<-lm(yExp ~ x, data = leastSquares)
summary(model3)
abline(model3)

#ejercicio 6 
n <- 20
M <- 5e3
mu<-0
sigma <- 1
beta_0 <- 1
beta_1 <- 2

x <- rchisq(n=n,df=5)

#Step2-4
beta_hat <- matrix(nrow = M,ncol=2)
linear_tren <- beta_0+beta_1*x

for (i in 1:M){
  #Step2: 
  eps <- rnorm(n=n,sd=sigma)
  
  #step3:
  y <- linear_tren +eps
  
  #Step4
  beta_hat[i,] <- coef(lm(y~x))
  
  
}

par(mfrow=c(1,2))
hist(beta_hat[,1],freq=F)
x<-seq(-0.5,2.5,len=100)
lines(x,dnorm(x,mean=mean(beta_hat[,1]),sd=sd(beta_hat[,1])),col='red',lty=2)
hist(beta_hat[,2],freq=F)
x<-seq(1.7,2.3,len=100)
lines(x,dnorm(x,mean=mean(beta_hat[,2]),sd=sd(beta_hat[,2])),col='red',lty=2)

error= rnorm(n=20,mu=0,sd=sigma)