task1_part2.rmd

(4) Another (and a more universal approach) to get rid of the dependence on $\theta$ in (2) is to estimate $s$ via the sample
standard error and use approximation of $\bar{X}$ via Student t-distribution; see details in Ross textbook on statistics or in the lecture notes
  
```{r}
for (n in c(100, 1000, 10000)) {
  cat("FOR SAMPLE SIZE = ", n, ":\n")
  
  sample_exp = rexp(n*m, lambda) # creating a single sample of exponential distribution
  sample_means = colMeans(matrix(sample_exp, nrow=n)) # creating a sample of sample means
  
  for (a in c(0.1, 0.05, 0.01)) { # iterating for each alpha
    cat("for alpha = ", a, ":\n")
    
    beta = 1 - a/2 
    St = qt(beta, n-1)
    
    prob = mean(abs(1 - sample_means / theta) <= St / sqrt(n)) # divided both sides of equation by theta
    cat("probability of theta being in the confidence interval = ", prob, "\n")
    
    interval_length = mean(sqrt(n)*sample_means*2*St / (n - St*St)) # long equation transformations that result into this
    cat("interval length = ", interval_length, "\n\n")
}
  cat("\n")
}
```