I am starting with Monte Carlo Simulation. I have run simulation to estimate the mean and the variance of the exponential distribution.

Simulation:

I have generated random sample from uniform distribution (sample size n) and then using the inverse function method generated the exponential distribution. I have run this simulation N1, N2, and N3 times where $N1<N2<N3$.

Results:

All worked well, when I plot the results I can see that the distribution of means and variances tends to the normal distribution as the number of simulation increases.

Questions:

1) Would you please help and clarify how the empirical and theoretical confidence intervals should be derived?

2) what is the manifestation of the Law of Large Numbers in this exercise?

3) and what is manifestation of Central Limit Theorem in this exercise?