# Code in Assignment #3 for STAT 460/560 

# Simulation study in Problem #2 

n_1 <- 100 # Prespecifed Sample size 
theta <- 3 # Prespecified mean of Poisson distribution 
eta <- exp(-theta) 

eps <- 0.01 # Prespecified value 

out_region <- numeric(1000) 

for (i in 1:1000){ 
  
  poisson_sample <- rpois(n = n_1, lambda = theta) 
  
  sample_sum <- sum(poisson_sample) 
  eta_hat <- (1 - 1/n_1)^(sample_sum) # The observed UMVUE for the parameter eta 
  
  out_region[i] <- (abs(eta_hat - eta) > eps) # Record whether the difference is larger than eps 
  
} 

# Calculate the simulation probability  

sum(out_region)/1000