var1 = function(x) {
  return(mean(x**2)-mean(x)**2)
}

var2 = function(x,n) {
  return(weighted.mean(x**2,n)-weighted.mean(x,n)**2)
}

sd1 = function(x) {
  return(sqrt(var1(x)))
}

sd2 = function(x,n) {
  return(sqrt(var2(x,n)))
}

covariance = function(x,y){
  return(sum((x-mean(x))*(y-mean(y)))/length(x))
}


# correlation = function(x, y) {
#   return(covariance(x,y)/(sd1(x) * sd1(y)))
# }

droite_yx = function(x, y) {
  m = covariance(x,y) / var1(x)
  p = mean(y) - m*mean(x)
  return (c(m, p))
}

droite_xy = function(x, y) {
  m = covariance(x, y) / var1(y)
  p = mean(x) - m * mean(y)
  return (c(m, p))
}

reg_yx <- droite_yx(x, y)
reg_xy <- droite_xy(x, y)

plot(x, y, 
     main = "Régression Vitesse vs Distance de freinage",
     xlab = "Vitesse (km/h)", 
     ylab = "Distance de freinage (m)",
     pch = 19, col = "blue", cex = 1.2)

abline(a = reg_yx[2], b = reg_yx[1], col = "red", lwd = 1)

abline(a = -reg_xy[2]/reg_xy[1], b = 1/reg_xy[1], col = "purple", lwd = 1)