#-----------------------------------------------------#
#---   Classical and Bayesian inference (AMS132)   ---#
#---   Class 13                                    ---#
#-----------------------------------------------------#

# Chi square distribution:
curve(dchisq(x, df=1), 0, 10, lwd=3)
curve(dchisq(x, df=2), 0, 10, lwd=3, add=TRUE, col="blue")
curve(dchisq(x, df=3), 0, 10, lwd=3, add=TRUE, col="red")
curve(dchisq(x, df=4), 0, 10, lwd=3, add=TRUE, col="green")
curve(dchisq(x, df=10), 0, 10, lwd=3, add=TRUE, col="magenta")



# Lactic acid concentration in cheese 
# a)
pchisq(10*0.3^2/0.09, df=10)

curve(dchisq(x, df=10), 0, 30, lwd=3)
abline( v=10*0.3^2/0.09, col="red", lwd=3)
polygon(c(0,seq(0,10*0.3^2/0.09, len=10),10), 
        c(0,dchisq(seq(0,10*0.3^2/0.09, len=10), df=10),0), col="blue")




# b)
n <- 10
sigma <- sqrt(0.09)
pnorm(0.1*sqrt(n)/sigma, 0, 1) - pnorm(-0.1*sqrt(n)/sigma, 0, 1)

curve(dnorm(x, 0, 1), -5,5, lwd=3)
abline(v=c(-0.1*sqrt(n)/sigma, 0.1*sqrt(n)/sigma), lwd=3, col="red")
sec <- seq(-0.1*sqrt(n)/sigma,0.1*sqrt(n)/sigma, len=10)
polygon(c(sec[1],sec,sec[10]), c(0,dnorm(sec, 0,1),0), col="blue")