exercise_2_a_iv.R

# Clear your environment of variables
rm(list = ls())
set.seed(1)
# We will use the rejection method to generate a discreet random variable
# X that takes values in 0,1,2,..., 10 with probabilities:

s = 0.6321311
p = c(s,s*exp(-1),s*exp(-2), s*exp(-3),s*exp(-4),s*exp(-5),s*exp(-6),s*exp(-7),s*exp(-8),s*exp(-9),s*exp(-10))

# We will do it from a Uniform discrete in {0,..,10} with mass q:

q =  replicate(11, 1/11)

# Find value of c such that c <= p(j)/q(j) for all j
# Setting it to the maximum of this ratio would work

c = max(p/q) # creates a vector p/q and takes its maximum


n = 11
x = c(0,1,2,3,4,5,6,7,8,9,10)

acceptance_ratio = exp(-x)


x_tuple = c()
counter_tuple = c()

for(i in 1:10000){
  flag = 1
  counter = 0
  
  while(flag){
    counter = counter + 1
    # First generate a Uniform discrete random variable in {0,1,2,...,10}
    # as we have done before:
    
    U1 = runif(1)
    Y = floor(n*U1) 
    
    # Generate another Uniform(0,1)
    U2 = runif(1)
    
    # Check if the condition is met
    
    K = p[Y+1]/(c*q[Y+1])
    
    # Alternatively we can use
    # K = acceptance_ratio[Y+1]
    
    if (U2 <= K) {
      X = Y
      flag = 0
    }
  }
  
  x_tuple = c(x_tuple,X)
  counter_tuple = c(counter_tuple,counter)
  
}

hist(x_tuple,c(0,1,2,3,4,5,6,7,8,9,10))

claim_amt = (300*x_tuple + 500)

hist(claim_amt,
     col = "darkblue",
     main = "Histogram for Team 1 Claim Amount",
     xlab = "Claim Amount",
     ylim = c(0,7000),
     border = "white",
     labels = TRUE)

# Exercise 2-A) iv)

# Hence, we see that the max frequency for our claim amount is 6391 
# which can be used to scale up our density curve, we use 6400 as an approximate guess

lines((300*x)+500, exp(-x)*6400, lwd = 3, col = "darkorange")


print(paste("The average number of iterations run = ",mean(counter_tuple)))
print(paste("The expected number of iterations expected(c) = ", c))