#-----------------------------------------------------#
#---   Classical and Bayesian inference (AMS132)   ---#
#---   Class 12                                    ---#
#-----------------------------------------------------#

#-- Bernoulli sampling

n <- 30

# distribution function of T:
tvalues <- 0:n
pfT <- dbinom(tvalues, n, 0.2)
plot(tvalues, pfT, type="h")

pfT <- dbinom(tvalues, n, 0.4)
plot(tvalues, pfT, type="h")

pfT <- dbinom(tvalues, n, 0.8)
plot(tvalues, pfT, type="h")


# P(T < 10):
probability  <- function(theta){
  t <- 0:9
  sum(choose(n, t)*theta^t*(1-theta)^(n-t))
}

theta <- seq(0, 1, length=100)
probs <- sapply(theta, probability)

plot(theta, probs, type="l")


# M.L.E:
n <- 10
theta <- seq(0.1, 0.9, by=0.1)
probs <- dbinom(10*theta, n, theta)

plot(theta, probs, type="p", lwd=3)


# Bayes estimate:
theta <- seq(0.1, 0.9, by=0.1)
probs <- pbinom(30*theta-7, n, theta)-pbinom(30*theta-13, n, theta)

points(theta, probs, col="blue", lwd=3)