### Ch. 8 Exercise 2

mu = 0.55

alpha = 1:2000
beta = alpha/mu - alpha

# Vector of probabilites for interval (0.51, 0.59)

prob = pbeta(0.59, alpha, beta) - pbeta(0.51, alpha, beta)
prob.err = abs(0.95 - prob)  # Errors for the probabilies

# Results: Traget parameter values
t.alpha = alpha[prob.err==min(prob.err)]
t.beta = round(t.alpha/mu - t.alpha)

t.alpha
t.beta

# Checking: Achieved mean and probability

a.mean = t.alpha/(t.alpha + t.beta)
a.mean
a.prob = pbeta(0.59, t.alpha, t.beta) - pbeta(0.51, t.alpha, t.beta)
a.prob

# Plot of the beta prior with the selected parameter values

x = seq(0,1,0.001)
y = dbeta(x,t.alpha,t.beta)

plot(x,y,type='l')